Resonance between planar self-affine measures
Full text
This is a sel -a chi ed e sion o an o iginal a icle. This e sion
may di e om he o iginal in pagina ion and ypog aphic de ails.
Au ho (s):
Ti le:
Yea :
Ve sion:
Copy igh :
Righ s:
Righ s u l:
Please ci e he o iginal e sion:
CC BY 4.0
h ps://c ea i ecommons.o g/licenses/by/4.0/
Resonance be ween plana sel -a ine measu es
© 2024 he Au ho s
Published e sion
Pyö älä, Aleksi
Pyö älä, A. (2024). Resonance be ween plana sel -a ine measu es. Ad ances in Ma hema ics,
451, A icle 109770. h ps://doi.o g/10.1016/j.aim.2024.109770
2024
Ad ances in Ma hema ics 451 (2024) 109770
Con en s lis s a ailable a ScienceDi ec
Ad ances in Ma hema ics
jou nal homepage: www.else ie .com/loca e/aim
Resonance be ween plana sel -affine measu es ✩
Aleksi Pyö älä
Depa men o Ma hema ics and S a is ics, P.O. Box 35 (MaD), FI-40014
Uni e si y o Jy äskylä, Finland
a i c l e i n o a b s a c
A icle his o y:
Recei ed 6 June 2023
Recei ed in e ised o m 27 May
2024
Accep ed 28 May 2024
A ailable online 14 June 2024
Communica ed by La s Olsen
MSC:
p ima y 28A80
seconda y 37A10
Keywo ds:
Sel -affine measu es
Hausdo ff dimension
Con olu ion o measu es
Resonance
We show ha i {ϕi}i∈Γand {ψj}j∈Λa e sel -affine i e a ed
unc ion sys ems on he plane ha sa is y s ong sepa a ion,
domina ion and i educibili y, hen o any associa ed sel -
affine measu es μand ν, he inequali y
dimH(μ∗ν)<min{2,dimHμ+dim
Hν}
implies ha he e is algeb aic esonance be ween he
eigen alues o he linea pa s o ϕiand ψj. This ex ends o
plana non-con o mal se ing he exis ing analogous esul s
o sel -con o mal measu es on he line.
© 2024 The Au ho (s). Published by Else ie Inc. This is an
open access a icle unde he CC BY license (h p://
c ea i ecommons .o g /licenses /by /4 .0/).
1. In oduc ion
In he 1960s, Fu s enbe g conjec u ed ha i X, Y⊆[0, 1] a e closed se s in a ian
unde mul iplica ion by in ege s mand n, espec i ely, hen o any s =0, he inequali y
✩The esea ch o his p ojec was conduc ed as pa o he au ho ’s doc o al s udies a Uni e si y o
Oulu, and has been pa ly suppo ed by he Resea ch Council o Finland ia he p ojec GeoQuan AM:
Geome ic and Quan i a i e Analysis on Me ic spaces, g an no. 354241. I hank Ville Suomala and Meng
Wu o hei commen s on an ea ly e sion o he pape , and A iel Rapapo and he anonymous e e ee
o many use ul commen s and sugges ions ha led o imp o ed p esen a ion o he pape .
E-mail add ess: aleksi. .pyo ala@jyu.fi.
h ps://doi.o g/10.1016/j.aim.2024.109770
0001-8708/© 2024 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he CC
BY license (h p://c ea i ecommons .o g /licenses /by /4 .0/).
2A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
dim(X+sY )<min{1,dim X+dimY}(1.1)
implies ha
log m
log n∈Q. He e and in he ollowing, dim deno es he (lowe ) Hausdo ff
dimension o bo h se s and measu es. This conjec u e was one o se e al ha aimed
o cap u e he idea ha i
log m
log n∈ Q, hen expansions in base mand nshould ha e
no common s uc u e: Indeed, he igh -hand side o (1.1)is always an uppe bound o
dim(X+sY ), while he s ic inequali y (1.1) implies ha many o he fibe s {(x, y) ∈X×
sY :x +y=z}a e la ge, which means ha Xand Yshould ha e a i hme ically simila
s uc u e in many places. The phenomenon (1.1)is usually e e ed o as esonance: X
and Ya e said o esona e i hey sa is y (1.1) o some s, and o he wise hey a e said
o dissona e.
I is also na u al o ask i a simila phenomenon holds in a mo e gene al se ing:
Fo dynamical sys ems Xand Y, does (1.1)imply some kind o algeb aic o a i hme ic
simila i y be ween he se s o he dynamics? The fi s esul in his di ec ion is due o
Mo ei a [33] om 1998, who p o ed ha o wo sel -con o mal se s on he line, (1.1)
canno hold in he p esence o an i a ionali y assump ion i one o he se s is o ally
non-linea . Recall ha a se K⊆Ris called sel -con o mal i
K=
m
i=1
i(K) (1.2)
o some C1+ε-con ac ions i.
In 2009, Pe es and Shme kin [36] es ablished analogous esul s o sums o sel -simila
se s on he line: The dimension o he sum is maximal unless he con ac ion a ios
o he defining simila i ies o m an a i hme ic se . A se A ⊆Ris called a i hme ic i
A ⊆αN o some α∈R. Recall ha a se is sel -simila i i sa isfies (1.2)wi h ibeing
simila i ies. An analogous esul o con olu ions o Can o measu es on he line was
ob ained sho ly a e by Naza o , Pe es and Shme kin [34]: Indeed, by eplacing sum
wi h con olu ion in (1.1), one can o mula e he concep o esonance o measu es.
A majo b eak h ough on he opic o esonance be ween dynamical sys ems was
achie ed by Hochman and Shme kin in 2012 [29], who in oduced a powe ul me hod
called he local en opy a e ages o a ack p oblems ega ding p ojec ions (and he e o e
sums) o dynamically defined ac als. Hochman and Shme kin managed o bo h p o e
he o iginal conjec u e o Fu s enbe g, and ex end o he se ing o measu es he exis ing
esul s on he sums o sel -simila and sel -con o mal se s on he line.
Namely, hey p o ed ha i { i}i∈Γand {gj}j∈Λa e amilies o C1+ε-con ac ions on
R ha sa is y he open se condi ion, hen o any associa ed sel -con o mal measu es μ
and ν,
dim(μ∗ν)=min{1,dim μ+dimν}(1.3)
unless he asymp o ic con ac ion a ios o φiand ψj o m an a i hme ic se . Recall ha
μis sel -con o mal i
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 3
μ=
i∈Γ
pi·μ◦ −1
i(1.4)
o a p obabili y ec o (pi)i∈Γand ias abo e. Due o well-known a ia ional p inciples,
he esul o sel -con o mal measu es implies he esul o se s as well. Recen ly, an
analogue o (1.3)was p o en by B uce and Jin [12] o a ich class o measu es on
homogeneous sel -simila se s wi hou equi ing any sepa a ion condi ions, in pa icula
measu es sa is ying (1.4)wi h ibeing simila i ies. Fo sel -con o mal measu es, (1.3)
was ecen ly e ified wi hou equi ing any sepa a ion condi ions by Bá ány, Käenmäki,
Wu and he au ho [5].
Pe haps su p isingly, almos no hing seems o be known o his phenomenon in highe
dimensions. While he e ce ainly exis s li e a u e on bounding he size o sums o con o-
lu ions om below, such as he amous in e se heo em o Hochman [26,27], i p ima ily
ocuses on showing ha , o e y gene al Xand Y, he sum (o con olu ion) X+Yis
s ic ly la ge han X, unless Xand Yha e a e y special s uc u e. See also [20,38]and
he e e ences he ein o p og ess in ela ed phenomena. Howe e , he exis ing esul s do
no aim o cap u e he spi i o he phenomenon p edic ed by Fu s enbe g, ha “geome -
ic esonance” o dynamically defined se s should imply a kind o “algeb aic esonance”
be ween he dynamics.
The pu pose o he p esen wo k is o p o ide an ex ension o his p inciple o he
plana se ing. Howe e , in o mula ing an ex ension beyond he line, one has o be
ca e ul: The di ec ex ension, ha
dim(X+Y)<min{2,dim X+dimY}(1.5)
unless he e is algeb aic esonance be ween he dynamics o Xand Y, b eaks down easily.
Indeed, one can o cou se isome ically embed any se s Xand Yon he line o he plane,
and hei sum will always ha e dimension a mos 1. I is also no difficul o cons uc
examples o Xand Ywi h dimension s ic ly g ea e han one by aking p oduc se s.
Thus, in o de o expec (1.5) o imply algeb aic esonance, one has o assume ha X
and Ya e “sp ead ou ” in sufficien ly many di ec ions, in some sense.
In his pape , we conside he size o μ ∗νwhen μand νa e sel -affine measu es
on he plane, ha is, hey sa is y (1.4)wi h ibeing in e ible affine con ac ions on
R2. Le RP1deno e he collec ion o one-dimensional subspaces o R2. Fo a 2 ×2-
ma ix A, le |λ1(A)| ≤|λ2(A)|deno e i s eigen alues. Le Aalso deno e he ac ion
induced by Aon RP1. We say ha an i e a ed unc ion sys em (an IFS o sho )
Φ ={ i(x) =Aix +ai}i∈Γo affine con ac ions on R2sa isfies
1) he s ong sepa a ion condi ion i he e exis s a bounded open se V=∅such ha
o e e y i =j∈Γ, i(cl(V)) ⊆Vand i(cl(V)) ∩ j(cl(V)) =∅,
2) hype bolici y i he e exis s n ∈Nand a wo d (i1, ..., in) ∈Γnsuch ha
|λ1(Ai1...A
in)| <|λ2(Ai1...A
in)|,
4A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
3) o al i educibili y i o e e y fini e se Θ ⊂RP1, he e exis s i ∈Γsuch ha
AiΘ =Θ, and
4) he domina ion condi ion i he e exis s a mul icone CRP1, i.e. a fini e union o
closed cones, such ha AiC⊆in (C) o each i ∈Γ.
Unde he domina ion condi ion, |λ1(Ai)| <|λ2(Ai)| o e e y i ∈Γby [4, Co olla y 2.4],
and o al i educibili y is equi alen o he a-p io i weake condi ion o i educibili y, ha
o e e y θ∈RP1 he e exis s i ∈Γsuch ha Aiθ=θ; see [7, Lemma 2.10].
Theo em 1.1. Le Φ ={ϕi(x) =Aix +ai}i∈Γand Ψ ={ψj(x) =Bjx +bj}j∈Λbe
sys ems o affine con ac ions on R2 ha sa is y he i educibili y, domina ion and s ong
sepa a ion condi ions. Le μand νbe ully suppo ed sel -affine measu es associa ed o
Φand Ψwi h dim μ ≥dim ν. I
dim(μ∗ν)<min{2,dim μ+dimν},
hen dim μ >1 >dim νand
{log |λ1(Ai)|:i∈Γ}∪{log |λ2(Bj)|:j∈Λ}
is an a i hme ic se .
Fo measu es which do no sa is y he domina ion condi ion, we ha e a pa ial esul
which holds unde a much weake sepa a ion assump ion: We say ha Φsa isfies he
exponen ial sepa a ion condi ion i he e exis s a cons an c >0and a le -in a ian
me ic don he g oup o in e ible affine maps on he plane such ha o any nand any
pai o wo ds (i1, ..., in) =(j1, ..., jn) ∈Γn, we ha e d(ϕi1◦···◦ϕin, ϕj1◦···◦ϕjn) >c
n.
Theo em 1.2. Le Φ ={ϕi(x) =Aix +ai}i∈Γand Ψ ={ψj(x) =Bjx +bj}j∈Λbe
sys ems o affine con ac ions on R2 ha sa is y he hype bolici y, o al i educibili y and
exponen ial sepa a ion condi ions. Le μand νbe ully suppo ed sel -affine measu es
associa ed o Φand Ψ. Then
dim(μ∗ν)≥min{1,dim μ}+min{1,dim ν}.
In pa icula , i dim(μ ∗ν) <min{2, dim μ +dimν}, hen dim μ >1 >dim νo
dim ν>1 >dim μ. In he p oo o Theo em 1.2 we do no con on he exponen ial sep-
a a ion condi ion di ec ly; We only use i o gain access o a esul o Bá ány, Hochman
and Rapapo [2]. We e e he eade o he pape s [2,28] o mo e discussion on he
exponen ial sepa a ion condi ion.
Le us now commen on he assump ions o Theo em 1.1. The assump ion o s ong
sepa a ion is classical in he s udy o i e a ed unc ion sys ems, since i makes i possible
o iew he a ac o as a dynamical sys em, gi ing access o a mul i ude o ools om
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 5
e godic heo y. Howe e , du ing ecen yea s, much a en ion has been di ec ed owa ds
es ablishing exis ing esul s wi hou assuming any sepa a ion condi ions, and we expec
i can be emo ed om ou esul as well.
The assump ion o hype bolici y ensu es ha he sys ems Φand Ψa e “s ic ly sel -
affine”. This is c ucial in ou app oach, since s ic ly sel -affine measu es ha e a e y
special angen ial s uc u e ha we will hea ily use. This s uc u e is o mula ed in
P oposi ion 4.1 which is he main echnical con ibu ion o his pape , and we belie e i
can find applica ions ou side his wo k as well. Wi hou he hype bolici y assump ion, he
con ac ions a e simila i ies up o a change o basis, meaning ha he sel -affine measu es
a e essen ially sel -simila . O cou se, i would be in e es ing o find an analogous esul
o esonance be ween plana sel -simila measu es.
The assump ion o o al i educibili y is ou way o ensu e ha he measu es μand ν
a e “sp ead ou ” in sufficien ly many di ec ions, which is some hing one has o assume as
explained in he p eceding discussion. Wi hou his assump ion, i is easy o cons uc
examples o which he conclusion o he heo em does no hold, by e.g. cons uc ing
measu es on Bed o d-McMullen ca pe s. Howe e , we do no know i i is enough o
assume o al i educibili y o jus one o he sys ems Φand Ψ.
I i happens ha dim μ ≥dim ν≥1o 1 ≥dim μ ≥dim ν, hen he s ong sepa a ion
(e en exponen ial sepa a ion), hype bolici y and o al i educibili y a e enough o ensu e
ha μ ∗νhas he maximal dimension, by Theo em 1.2. The case dim μ >1 >dim ν
is mo e delica e, since now μmigh be la ge enough o “abso b” some o νin he con-
olu ion. Howe e , i he a i hme ic conclusion o Theo em 1.1 does no hold, hen we
can use he domina ion condi ion o show ha he measu es μand νa e “spaced” in
such a manne ha his kind o abso p ion canno happen. Howe e , i is likely ha he
equi emen o domina ion is jus a by-p oduc o ou me hod.
Finally, we ema k ha Theo ems 1.1 and 1.2 combined wi h he a ia ional p inciple
o Bá ány, Käenmäki and Rossi [6, P oposi ion 2.4] yield an analogous esul o se s:
Co olla y 1.3. Le Φ ={ϕi(x) =Aix +ai}i∈Γand Ψ ={ψj(x) =Bjx +bj}j∈Λbe
sys ems o affine con ac ions on R2 ha sa is y he domina ion, i educibili y, and s ong
sepa a ion condi ions. Le Xand Ydeno e he a ac o s o Φand Ψ, and suppose ha
dim X≥dim Y. I
dim(X+Y)<min{2,dim X+dimY},
hen dim X>1 >dim Yand he e exis s n ∈Nsuch ha
{log |λ1(Ai1...A
in)|:(i1,...,i
n)∈Γn}∪{log |λ2(Bj1...B
jn)|:(j1,...,j
n)∈Λn}
is an a i hme ic se .
P oo . Suppose ha dim(X+Y) <min{2, dim X+dimY}and le ε >0be small.
Using [6, P oposi ion 2.4], choose n ∈Nand ully suppo ed sel -affine measu es μand
6A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
νassocia ed o he sys ems {ϕi1◦···◦ϕin}(i1,...,in)∈Γnand {ψj1◦···◦ψjn}(j1,...,jn)∈Λn,
such ha dim μ ≥dim X−εand dim ν≥dim Y−ε. Then, since μ ∗νis suppo ed on
X+Y, o small enough εwe ha e
dim(μ∗ν)≤dim(X+Y)<min{2,dim X+dimY}−3ε≤min{2,dim μ+dimν}−ε.
The conclusion ega ding eigen alues now ollows om Theo em 1.1. In addi ion, we
ha e dim μ >1 >dim ν, which implies ha dim X>1 ≥dim Yi εwas chosen small
enough. I i we e ha dim Y=1, hen dim μ >1 ≥dim ν≥1 −ε, and Theo em 1.2
would asse ha dim(μ ∗ν) >2 −εwhich is a con adic ion. Thus dim Y<1and he
p oo is comple e.
1.1. On he p oo s o Theo ems 1.1 and 1.2
We begin wi h he p oo o Theo em 1.2. Ou p oo is based on he local en opy
a e ages o Hochman-Shme kin [29]: ins ead o p o ing di ec ly ha dim(μ ∗ν)is la ge,
we will show ha he a e age o he fini e-scale en opies o μk∗νko e many k∈N
is la ge, whe e μkand νka e magnifica ions (a scale 2−k) o μand νalong p ope ly
chosen fil a ions o hei suppo s.
Indeed, magni ying bo h μand νalong he “cylinde ellipses” ϕi1◦···◦ϕin(B(0, 1))
and ψj1◦···◦ψjn(B(0, 1)), on he limi hey bo h esemble o hogonal p ojec ions o
he o iginal measu es, by he hype bolici y assump ion. Relying on he exponen ial sep-
a a ion condi ion and he s ong p ojec ion heo em o sel -affine measu es [2, Theo em
1.3] due o Bá ány, Hochman and Rapapo , hese magnifica ions will ha e en opy
close o min{1, dim μ}and min{1, dim ν}, espec i ely. Since he i educibili y assump-
ion ensu es ha he ellipses on which he magnifica ions a e suppo ed on ha e majo
semi-axes poin ing in diffe en di ec ions, hei con olu ion has a p oduc -like s uc u e
on he plane and hus has en opy close o min{1, dim μ} +min{1, dim ν}. This essen ially
concludes he p oo o Theo em 1.2.
I is a o mal consequence o Theo em 1.2 ha o sel -affine measu es μand νas in
he s a emen wi h dim μ ≥dim ν, he inequali y dim(μ ∗ν) <min{2, dim μ +dimν}
is only possible when dim μ >1 >dim ν. This is he fi s s a emen o Theo em 1.1.
Gi en measu es μand νwi h dim μ >1 >dim ν, we show ha in he case whe e
{log |λ1(Ai)| :i ∈Γ} ∪{log |λ2(Bj)| :j∈Λ}is no an a i hme ic se , we will in
ac ha e dim(μ ∗ν) =min{2, dim μ +dimν}, con adic ing he assump ion o s ic
inequali y.
As in he p oo o Theo em 1.2, we ely on he local en opy a e ages. Howe e , a
consequence o he assump ion dim μ >1 >dim νis ha he magnifica ions o μalong
he cylinde ellipses can no longe s o e a sufficien amoun o he dimension o μ o us
o be able o each he desi ed lowe bound o he dimension o μ ∗ν. Ins ead, we will
magni y μalong dyadic squa es and νalong he cylinde ellipses.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 7
Fo any such magnifica ions μkand νko μand ν, and any o hogonal p ojec ion π,
applying he chain ule o en opy yields ha he en opy o μk∗νkis equal o
en opy o πμk∗πνk+ condi ional en opy o μk∗νkw. . . π
which in u n is bounded om below by
en opy o πμk∗πνk+ en opy o μk−en opy o πμk,(1.6)
using again he chain ule and he ac ha con olu ion ne e dec eases en opy. This
obse a ion is pu o use h ough ou key geome ic ing edien : We show ha o nea ly
e e y k, he magnifica ion μkhas a fibe s uc u e in he sense ha πμkis (close o)
a slice measu e o μ, o a p ope ly chosen π. The p ecise o m o his s uc u e is
s a ed in P oposi ion 4.1. Simila fibe s uc u es ha e been p e iously obse ed o
sel -affine se s by Käenmäki, Koi usalo and Rossi [30], o sel -affine measu es on Bed o d-
McMullen ca pe s by Fe guson, F ase and Sahls en [19]and o sel -affine measu es
wi h an addi ional p ojec ion condi ion by Kemp on [32]. Combining his wi h he di-
mension conse a ion phenomenon ha ollows o plana sel -affine measu es om he
Led appie -Young o mula o Bá ány [1]and he gene al ac ha when a e aged o e
many k, he en opy o μkis close o he dimension o μ, we see ha upon a e aging,
(1.6)is in ac close o
en opy o πμk∗πνk+1,(1.7)
ecalling ha we a e assuming dim μ >1.
Thus, i we manage o show ha o mos k, o he measu es πμkand πνkon he
line we ha e
en opy o πμk∗πνk≥min{1,en opy o πμk+ en opy o πνk}−o(1),(1.8)
whe e o(1) deno es a quan i y ha anishes as he scale o he en opy dec eases, hen
we ha e ha he en opy o μk∗νkwhen a e aged o e many kis a leas
min{2,en opy o πμk+ 1 + en opy o πνk}−o(1)
=min{2,dim μ+dimν}−o(1)
by (1.7), ano he applica ion o he dimension conse a ion o [1]and he p ojec ion
heo em o [2]. This would conclude he p oo .
P o ing (1.8)is he pa whe e he assump ion on he eigen alues o Aiand Bjs eps
in, and whe e he domina ion condi ion is mos hea ily u ilized. Acco ding o he classi-
cal p ojec ion heo em o Ma s and (combined wi h some known exac -dimensionali y
esul s), (1.8)would hold i we we e allowed o scale ei he o he measu es πμko πνk
by a andom eal numbe . Because o his, i we we e able o show ha he sequence
8A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
(πμk×πνk)k∈Nequidis ibu es o a dis ibu ion on he space o measu es which is
join ly in a ian unde scaling in ho izon al and e ical di ec ions, (1.8)would ollow
om an applica ion o Fubini and Ma s and’s heo em.
The fi s s ep in es ablishing he equidis ibu ion o (πμk×πνk)k∈Nis he obse a ion
ha since πμkis (close o) a slice o μand πνkis an o hogonal p ojec ion o ν, hey bo h
can be exp essed as images o poin s in ce ain unde lying con inuous- ime dynamical
sys ems ZΦ=(ZΦ, {T } ∈R, λΦ)and ZΨ=(ZΨ, {S } ∈R, λΨ) defined in Sec ion 6. As in
[12,29], his allows us o desc ibe he sequence (πμk×πνk)k∈Nas an image o an o bi o
a poin in he p oduc sys em ZΦ×Z
Ψ, and so he s udy o equidis ibu ion o (πμk×
πνk)k∈Nin he space o p obabili y measu es is educed o he s udy o equidis ibu ion
o o bi s in ZΦ×ZΨ. In ou se ing, he s udy o equidis ibu ion p ope ies o ZΦ×ZΨ
equi es me hods which a e e y diffe en om hose o [12,29], whe e he sys ems Φ
and Ψwe e assumed o be sel -simila , and which, o ou knowledge, ha e no appea ed
be o e in a ac al-geome ic se ing.
In he p oo o P oposi ion 6.1, we disco e a ela ion be ween he eigen alues o
he sys ems ZΦand ZΨand he eigen alues o he linea pa s o he unc ions in Φ
and Ψ. Using his ela ion we find ou ha unde he assump ion ha {log |λ1(Ai)| :
i ∈Γ} ∪{log |λ2(Bj)| :j∈Λ}is no an a i hme ic se , he flows ZΦand ZΨha e no
common eigen alues and so, applying me hods om e godic heo y, we show ha ypical
o bi s in ZΦ×ZΨwill equidis ibu e o he p oduc measu e λΦ×λΨ. Consequen ly, he
sequence (πμk×πνk)k∈Ncan be exp essed as an image o an o bi which equdis ibu es
o λΦ×λΨ, and so he sequence (πμk×πνk)k∈Nwill equidis ibu e o a dis ibu ion
wi h he desi ed join in a iance unde scaling in e ical and ho izon al di ec ions.
1.2. S uc u e
In Sec ion 2we in oduce ou se ing mo e igo ously, and collec some gene al known
esul s and sho lemmas on sel -affine measu es, dynamical sys ems and andom ma ix
p oduc s. Sec ion 3is de o ed o ansla ing he local en opy a e ages machine y o [29]
o ou se ing, while in Sec ion 4we s a e ou main echnical esul s, and use hese o
conclude he p oo o Theo em 1.1. Sec ion 5is pe haps he mos echnical one, de o ed
o he p oo o he main geome ic esul , P oposi ion 4.1. The a gumen s he e we e
inspi ed by he wo k o Kemp on [32]. Finally, in Sec ion 6we in es iga e he dynamics
o he sequences o magnifica ions o μand ν, and p o e he equi ed lowe bounds o
hei a e age en opies. The no a ion used in he pape is summa ized in Table 1.
2. P elimina ies
In his pape , a measu e e e s o a Radon measu e on a me izable opological space.
The no a ion P(X) s ands o p obabili y measu es on he space X. Fo a measu e μon
Xand a subse Y⊆X, μ|Ydeno es he es ic ion o μon o Y, μY:= μ(Y)−1μ|Y he
no malized es ic ion when μ(Y) >0. Fo a measu able unc ion , le μ := μ ◦ −1
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 15
2.5. Flows and eigen alues
In he ollowing, le Xbe a comple e sepa able me ic space and le μbe a Bo el
p obabili y measu e on X. Le {T :X→X} ∈Rbe a amily o measu able unc ions
wi h he p ope y ha T0=Idand Ts◦T =Ts+ o each s, ∈R. Recall ha a eal
numbe c ∈Ris an eigen alue o he flow (X, {T } ∈R, μ)i he e exis s a measu able
unc ion :X→Csuch ha ≡ 0and ◦T (x) =e(c ) (x) o μ-almos e e y
xand e e y ∈R, whe e we w i e e(x) := exp(2πix). Such a unc ion is called an
eigen unc ion o c. Recall ha he flow (X, {T } ∈R, μ)is e godic i and only i 0is a
simple eigen alue o (X, {T } ∈R, μ), ha is, e e y in a ian unc ion is cons an almos
e e ywhe e. We now eco d some s anda d p ope ies o eigen alues.
Lemma 2.8. Suppose ha T :X→Xis con inuous o each ∈R, and ha
(X, {T } ∈R, μ)is e godic. Fo any c ∈R, he disc e e- ime dynamical sys em (X, Tc, μ)
is e godic i and only i no non-ze o in ege mul iple o c−1is an eigen alue o
(X, {T } ∈R, μ).
P oo . This is p o ed in [24, Lemma 3.11] in a sligh ly diffe en language. Fo he con-
enience o he eade , we epea he sho p oo he e.
Le μ =´μxdμ(x)be he e godic decomposi ion o μwi h espec o he map Tc.
F om {T } ∈R-in a iance o μi ollows ha μ =´c
0T μ d =˜c
0T μxd dμ(x). Since
he unc ion x → ´c
0T μxd is {T } ∈R-in a ian , i is cons an almos e e ywhe e and
so we ha e μ =´c
0T μxd o μ-almos e e y x. Fix now such an x, le ν:= μxand
no e ha he unc ion F:R/cZ →P(X)gi en by F( ) =T νis well-defined since
Tcν=ν. Le Λ ={ ≥0 :F( ) =F(0)}deno e he se o pe iods o F. Since Fis
Lebesgue measu able (e en con inuous, see e.g. [15, P oposi ion 3.1]), ei he Λ =R/cZo
Λ ={kc/n :0 ≤k≤n} o some n ∈N. In he o me case, μ =´c
0F( ) d =F(0) =ν
and so μis Tc-e godic. In he la e case, he e exis s a la ges n ∈Nsuch ha Tc/nν=ν
and so μ =´c/n
0T νd . In pa icula , n/c is an eigen alue o (X, {T } ∈R, μ) o he
eigen unc ion :X→Cdefined by (x) =e( (x)n/c) o μ-almos e e y x, whe e
(x) ∈R/cn−1Zis he unique numbe such ha limk→∞ 1
kk
=1 δT
cx=T (x)ν.
Lemma 2.9. The flow (X, {T } ∈R, μ)has a mos coun ably many eigen alues.
P oo . I is no difficul o see ha eigen unc ions o diffe en eigen alues a e o hogonal.
Since he space L2(X)is sepa able when Xis comple e and sepa able, any collec ion o
o hogonal unc ions has o be coun able.
The ollowing cha ac e iza ion o e godici y o a p oduc o e godic sys ems plays a
key ole in he p oo o Theo em 1.1.
P oposi ion 2.10. Le (X, {T } ∈R, μ)and (Y, {S } ∈R, ν)be e godic flows o e comple e
sepa able me ic measu e spaces (X, μ)and (Y, ν). The p oduc flow (X×Y, {T ×
16 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
S } ∈R, μ ×ν)is e godic i and only i he flows ha e no common eigen alues o he
han 0.
A de ailed p oo o he p oduc o disc e e- ime dynamical sys ems can be ound
o example in [25, Theo em 6.6.6]. While we belie e ha he esul o he p oduc o
e godic flows is also well-known, we we e unable o find a e e ence in he li e a u e and
so p o ide a sho p oo elying on he spec al heo em o uni a y ope a o s. The o m
o he spec al heo em we equi e is eco ded below.
Theo em 2.11 (Spec al heo em o uni a y ope a o s). Le (U ) ∈Rbe a s ongly con-
inuous one-pa ame e uni a y g oup on a Hilbe space H. Then he e exis s a fini e
measu e ηon R ×Nand a uni a y map R:L2(R ×N, η) →Hsuch ha
U =RL R−1,
o e e y ∈R, whe e he ope a o L :L2(R ×N, η) →L2(R ×N, η)is defined by
L (a(x, n))(x, n) =e( x)a(x, n) o e e y a ∈L2(R ×N, η)and η-almos e e y (x, n).
P oo . By S one’s heo em o one-pa ame e uni a y g oups [22, Theo em 10.15], he e
exis s a sel -adjoin ope a o Asuch ha U =e( A) =∞
k=0(2πi A)k/k! o e e y ∈R.
Applying he spec al heo em o sel -adjoin ope a o s [14, Theo em 2.5.1] we find a
fini e measu e ηon R ×Nand a uni a y map R:L2(R ×N, η) →Hsuch ha e( A) =
Re( h)R−1, whe e h ∈L2(R ×N, η)is he map h(x, n) =x. Se ing L =e( h) o e e y
∈R, we ha e U =RL R−1and L (a)(x, n) =e( h(x, n))a(x, n) =e( x)a(x, n) o
e e y a ∈L2(R ×N, η)and η-almos e e y (x, n), as was equi ed.
P oo o P oposi ion 2.10.Le be an in a ian unc ion o {T ×S } ∈R. I suffices
o show ha is cons an μ ×ν-almos e e ywhe e. Le
˜
T and
˜
S deno e he linea
Koopman ope a o s on he Hilbe spaces L2(X, μ)and L2(Y, ν) defined by
˜
T g=g◦T
and
˜
S g=g◦S . Since T and S a e measu e-p ese ing, hese ope a o s a e easily seen
o be uni a y. Mo eo e , he g oups (˜
T ) ∈Rand (˜
S ) ∈Ra e s ongly con inuous, see
o example [15, P oposi ion 3.1]. Le {ϕi}i∈Nand {ψj}j∈Nbe o hono mal bases o
L2(X, μ)and L2(Y, ν). Then {φiψj}i,j∈N o ms an o hono mal basis o L2(X×Y, μ ×ν),
so we can expand =i,j ai,jϕiψj o some ai,j ∈C. Decomposing
˜
T =RTLT
R−1
T
and
˜
S =RSLS
R−1
Susing Theo em 2.11, we may u he decompose
˜
T ⊗˜
S =(RT⊗RS)(LT
⊗LS
)(R−1
T⊗R−1
S).(2.2)
Using (2.2)and in a iance o , we ha e
(R−1
T⊗R−1
S) =LT
⊗LS
(R−1
T⊗R−1
S)
=
i,j
ai,je( (x+y))R−1
T⊗R−1
Sϕiψj.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 17
By uniqueness o coo dina e ep esen a ion, o any (i, j)such ha ai,j =0we ha e
R−1
TϕiR−1
Sψj=e( (x +y))R−1
TϕiR−1
Sψj, in pa icula , R−1
Tϕi(x, n)R−1
Sψj(y, m) =0
whene e y=−x. Fixing any such (i, j), he e exis s c ∈Rsuch ha R−1
Tϕi(x, n) =0
whene e x =cand R−1
Sψj(y, m) = 0 whene e y=−c. I easily ollows ha cis an
eigen alue o bo h {T } ∈Rand {S } ∈R o he eigen unc ions ϕiand ψj:
ϕi◦T =RTLT
R−1
Tϕi≡RTe(c )R−1
Tϕi=e(c )ϕi
and simila ly, ψj◦S ≡e(c )ψj. Since {T } ∈Rand {S } ∈Rha e no common eigen alues
o he han 0, we conclude ha c =0. Bu his implies ha ϕiand ψja e in a ian
unc ions o {T } ∈Rand {S } ∈Rand he e o e cons an almos e e ywhe e. Since his
is ue o any pai (i, j)such ha ai,j =0in he ep esen a ion =i,j ai,jϕiψj, we
conclude ha also is cons an almos e e ywhe e.
Le (ΓN, σ)be a shi space. Gi en a unc ion :Γ
N→(0, +∞), we may build a semi-
flow om (ΓN, σ)by “flowing up” om a poin i ∈ΓNun il we each he ime (i), hen
swi ch o he poin σiand con inue flowing un il he ime (σi), and so on. Fo mally,
we le Z=(Γ
N×[0, ∞))/ ∼equipped wi h he quo ien opology, whe e ∼deno es he
equi alence ela ion gene a ed by (i, (i)) ∼(σi, 0) o e e y i ∈ΓN. Deno ing each
equi alence class [(i, )] by he unique ep esen a i e (i, )wi h 0 ≤ < (i), we may
w i e
Z={(i, ): i∈ΓN,0≤ ≤ (i)}.
Le Ts:(i, ) → (i, +s) o e e y s ≥0. I he e is a σ-in a ian measu e μon ΓN
which measu es and ´ dμ <∞, a na u al {Ts}s≥0-in a ian p obabili y measu e
on Zis gi en by λ := (μ ×L)Z. The ipe (ΓN, {Ts}s≥0, λ)is called he suspension o
(ΓN, σ, μ) unde he oo unc ion . I is easy o see ha i (ΓN, σ, μ)is e godic, hen so is
(Z, {Ts}s≥0, λ): Fo i ϕwe e a non-cons an in a ian unc ion o (Z, {Ts}s≥0, λ), hen
i → ϕ(i, 0) would be a non-cons an in a ian unc ion o (ΓN, σ, μ). Suspension flows
o e he in e ible sys em (ΓZ, σ)a e defined analogously by se ing Z=(Γ
Z×R)/ ∼,
whe e ∼is he equi alence ela ion gene a ed by (i, ) ∼(σi, − (i)) o e e y (i, ) ∈
ΓN×R, and in his case we le Ts:(i, ) → (i, +s) o e e y s ∈R.
A special p ope y o egula enough suspension flows is ha o any eigen alue co -
esponds a con inuous eigen unc ion.
P oposi ion 2.12 (P oposi ion 6.2 o [35]). Le (Z, {Ts}s≥0, λ)be he suspension o a
shi space (ΓN, σ, μ)unde a locally Hölde con inuous oo unc ion, whe e μis he
equilib ium s a e o a locally Hölde con inuous po en ial on ΓN. Then a numbe α∈R
is an eigen alue o (Z, {Ts}s≥0, λ) o a con inuous eigen unc ion i and only i i is an
eigen alue o a measu able eigen unc ion.
18 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
I is well-known ha Be noulli measu es on ΓNa e equilib ium s a es o locally
cons an po en ials.
Lemma 2.13. Le :Γ
Z→(0, +∞)be a con inuous unc ion such ha (i) = (j)
whene e i+=j+, and le +:Γ
N→[0, +∞)deno e he unc ion gi en by +(i) =
(j) o e e y i ∈ΓNand j ∈ΓZsuch ha j+=i.
Le μbe a shi -in a ian e godic measu e on ΓZwhich measu es , and w i e μ+ o
he p ojec ion on o he posi i e coo dina es. Le (Z, {Ts}s≥0, λ) =: Zdeno e he suspen-
sion o (ΓZ, σ, μ)o e , and (Z+, {Ts}s≥0, λ+) =: Z+ he suspension o (ΓN, σ, μ+)
o e +.
Then o any eigen alue α=0o Z, he e exis s n ∈Zsuch ha nα is an eigen alue
o Z+.
P oo . Le α=0be such ha nα is no an eigen alue o Z+ o any n ∈Z. Then by
Lemma 2.8, he disc e e- ime sys em (Z+, T1/α, λ+)is e godic. (By possibly eplacing α
by −α, we may assume α>0.) Ou aim is o show ha also he sys em (Z, T1/α, λ)is
e godic. Le U⊆Zbe an open se o he o m [i]n
m×I o some i ∈ΓZ, m ≤n ∈Zand
an in e al I⊆R. Since any con inuous unc ion on Zcan be app oxima ed a bi a ily
well (in L1) by simple unc ions on his kind o se s, in o de o show ha λis e godic
unde T1/α, i suffices o show ha
lim
n→∞
1
n#{0≤k≤n:Tk/α(i, )∈U}=λ(U)
o a.e. (i, ).
Le ∈Nbe la ge enough so ha o e e y (i, ) ∈T−/αU, also (j, ) ∈T−/αU
whene e j+=i+. W i e (T−/αU)+ o he p ojec ion o T−/αUon o Z+, and no e
ha T−/αUequals he embedding o (T−/αU)+ o Z.
Le A+be he se o ull λ+-measu e such ha o each (j, ) ∈A+, we ha e
lim
n→∞
1
n#{0≤k≤n:Tk/α(j, )∈(T−/αU)+}=λ+((T−/αU)+)=λ(T−/αU)=λ(U),
by Bi khoff’s e godic heo em. The second- o-las equali y ollows om he choice o ,
and he las ollows om T -in a iance o λ. Now, i Ais he embedding o A+ o Z,
hen T/αAhas ull λ-measu e, and o each (i, ) ∈T/αA,
lim
n→∞
1
n#{0≤k≤n:Tk/α(i, )∈U}
= lim
n→∞
1
n#{0≤k≤n:T(k−)/α(i, )∈T−/αU}
=λ(U).
The e o e λis e godic unde T1/α and by Lemma 2.8, αis no an eigen alue o Z.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 19
2.6. Shannon en opy
Fo a p obabili y measu e μon Rdand any measu able pa i ions Eand Fo Rd, we
w i e H(μ, E) =−
E∈E μ(E) log μ(E) o he (Shannon) en opy o μwi h espec o
E, and H(μ, E|F) =−
F∈F μ(F)H(μF, E) o he condi ional en opy o μwi h espec
o E, gi en F. W i e E∨F={E∩F:E∈E, F∈F} o he join o Eand F. In he
ollowing, we eco d some elemen a y p ope ies o en opy. We e e o [13, Sec ion 2]
and [40, Sec ions 4.2 and 4.3] o mo e de ailed discussions on he opic.
Lemma 2.14 (Chain ule, Theo em 4.3 o [40]). Le μbe a p obabili y measu e on R2,
and le E, Fbe pa i ions o R2. Then
H(μ, E∨F)=H(μ, E|F)+H(μ, F).
Lemma 2.15 (Conca i y and almos -con exi y). Fo any pa i ions Eand Fo R2, he
unc ion μ → H(μ, E|F)is conca e. Mo eo e , he unc ion μ → H(μ, E)is almos
con ex in he sense ha o any p obabili y measu es μ1, ..., μkand a p obabili y ec o
(p1, ..., pk),
Hk
i=1
piμi,E≤
k
i=1
piH(μi,E)−
k
i=1
pilog pi.
P oo . The conca i y o he unc ion μ → H(μ, E|F) ollows essen ially om he con-
ca i y o x →−x log xon [0, 1]; Fo de ails, we e e o [13, Equa ions (2.29) and (2.43)
and Theo ems 2.7.2 and 2.7.3]. The second claim is immedia e om he defini ion o
en opy:
Hk
i=1
piμi,E=−
k
i=1
pi
E∈E
μi(E)logk
i=1
piμi(E)
≤−
k
i=1
pi
E∈E
μi(E)(log pi+logμi(E))
=
k
i=1
piH(μi,E)−
k
i=1
pilog pi.
Lemma 2.16. Le μbe a p obabili y measu e on Rd, and le Eand Fbe pa i ions o Rd
such ha each elemen o Ein e sec s a mos kelemen s o Fand ice e sa. Then
|H(μ, E)−H(μ, F)|≤log k.
P oo . We fi s no e ha H(μ, E∨F) ≥H(μ, E)by [40, Theo em 4.3]. Lemma 2.14 now
asse s ha H(μ, E) −H(μ, F) ≤H(μ, E∨F) −H(μ, F) =H(μ, E|F)and since o each
20 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
F∈F, μFis suppo ed on a mos ka oms o E, i ollows om [40, Co olla y 4.21]
ha H(μ, E|F) ≤log k. Simila ly, H(μ, F) −H(μ, E) ≤log k.
A consequence o conca i y is ha aking a con olu ion can dec ease en opy a mos
by an addi i e cons an .
Lemma 2.17. Le Eand Fbe pa i ions o R2such ha any ansla e o E∨F in e sec s
a mos kelemen s o E∨F and any ansla e o Fin e sec s a mos kelemen s o F.
Then o any p obabili y measu es μand ν,
H(μ∗ν, E|F)≥H(μ, E|F)−2logk.
P oo . I ollows om Lemma 2.15 and Jensen’s inequali y ha H(μ ∗ν, E|F) ≥´H(μ ∗
δx, E|F) dν(x). By Lemmas 2.14 and 2.16, H(μ ∗δx, E|F) ≥H(μ, E|F) −2 log k o e e y
x ∈R2.
Le Dn=Dn(Rd)deno e he pa i ion o Rdin o dyadic cubes o side-leng h 2−n
which we call he “le el-n” dyadic pa i ion. Fo x ∈Rd, w i e Dn(x) o he elemen o
Dn ha con ains x. Fo en opy wi h espec o he dyadic pa i ion we use he sho -
hand no a ion Hn(μ) =H(μ, Dn) =−
D∈Dnμ(D) log μ(D). Fo θ∈RP 1, le Dn(θ)
deno e he le el-ndyadic pa i ion o he line θ. The ollowing is a simple applica ion
o he chain ule; Le Rθdeno e he “sho es ” o a ion which akes θ∈RP1on o he
y-axis, wi h Rx-axis gi en by he clockwise o a ion.
Lemma 2.18. Le μbe a p obabili y measu e on R2, and le θ∈RP1. Then, deno ing
Hn(μ|πθ) := H(μ, Dn(R2)|π−1
θDn(θ)), we ha e
|Hn(μ)−(Hn(πθμ)+Hn(μ|πθ))|≤log 9
o e e y n ∈N.
P oo . I ollows om Lemma 2.14 ha H(μ, Dn(R2) ∨π−1
θDn(θ)) =H(πθμ, Dn(θ)) +
Hn(μ|πθ). I is no difficul o see ha each elemen o Dn(R2) in e sec s a mos wo
elemen s o Dn(R2) ∨π−1
θDn(θ)and ice e sa. On he o he hand, each elemen o
Dn(θ) in e sec s a mos h ee elemen s o Dn(R2)and ice e sa, whence i ollows
om Lemma 2.16 ha |Hn(μ) −(Hn(πθμ) +Hn(μ|πθ))| ≤log 9.
Lemma 2.19. Le μand νbe p obabili y measu es on [0, 1], and suppose ha μis non-
a omic. Then o e e y >0and ε >0, he e exis s N0∈Nsuch ha o any in e al
Iwi h μ(I) ≥ , we ha e
1
NHN(μI∗ν)≥dim(μ∗ν)−ε
o e e y N≥N0.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 21
P oo . Le K ={(a, b) ∈R2:μ([a, b]) ≥ }. We will show ha o e e y >0such
ha K is nonemp y,
1
NHN(μ[a,b]∗ν)is con inuous in (a, b) ∈K (in he subspace
opology) and he con inui y is uni o m in N.
Le , ε >0be gi en, and le δ>0be small wi h espec o εand . Fix (a0, b0) ∈K
and le Iδbe he la ges in e al con ained in [a, b] o e e y (a, b) ∈B((a0, b0), δ). I
δis small enough, we ha e
μ(Iδ)
μ([a,b]) ∈[1 −ε, 1 +ε] o e e y (a, b) ∈B((a0, b0), δ), by
non-a omici y o μand he assump ion μ([a0, b0]) ≥ .
Now, o e e y (a, b), (a, b) ∈B((a0, b0), δ), applying bilinea i y o con olu ion and
Lemma 2.15 in he fi s and second- o-las inequali ies, we ha e
HN(μ[a,b]∗ν)
≤μ(Iδ)
μ([a, b])HN(μIδ∗ν)+1−μ(Iδ)
μ([a, b])HN(μ[a,b] Iδ∗ν)+ε
≤(1 + ε)μ(Iδ)
μ([a,b
])HN(μIδ∗ν)+2Nε
≤(1 + ε)HN(μ[a,b]∗ν)+2Nε
≤HN(μ[a,b]∗ν)+3Nε.
Thus (a, b) → 1
NHN(μ[a,b]∗ν)is con inuous in K , uni o mly in N. On he o he hand,
o e e y (a, b) ∈K ,
dim(μ∗ν)≤dim(μ[a,b]∗ν)≤lim in
N→∞
1
NHN(μ[a,b]∗ν)
by [17, Theo em 1.3]. By uni o m con inui y, his con e gence is uni o m in K , which
is wha we wan ed o p o e.
2.7. Magni ying measu es
Fo x ∈Rdand ≥0, we le Tx:y→ y−xdeno e he ansla ion aking x o he
o igin, and S :x → 2 x he exponen ial “magnifica ion” ope a ion. We le S∗
deno e
he ac ion o S on measu es equipped wi h no maliza ion and es ic ion, ha is,
S∗
μ(A)=μ(B(0,2− ))−1μ(2− A∩B(0,2− ))
o e e y Bo el se A ⊆B(0, 1) and measu e μwhose suppo con ains he o igin.
The e is a na u al way in which measu es on Rdgi e ise o measu es on P(Rd).
Namely, conside he sequence (S∗
Txμ) ≥0, called he scene y o μa x. The s a is ical
p ope ies o his sequence a e desc ibed by he accumula ion poin s o he sequence
1
´
0δS∗
Txμd ≥1, called he scene y flow o μa x. The accumula ion poin s o he
scene y flow in he weak-∗ opology a e measu es on P(Rd), and a e called angen
22 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
dis ibu ions o μa x. The measu e μis called uni o mly scaling i he scene y flow
con e ges almos e e ywhe e o a unique angen dis ibu ion P. I is hen said ha μ
gene a es P.
A ema kable esul o Hochman [23]is ha angen dis ibu ions a almos e e y
poin a e ac al dis ibu ions, objec s which enjoy s ong spa ial in a iance p ope ies.
We gi e he defini ion he e o comple eness, al hough we will no use i di ec ly.
Defini ion 2.20. An S∗
-in a ian measu e Pon P(Rd)is called a ac al dis ibu ion i
o any measu able A, P(A) =1i and only i o e e y >0, P-almos e e y νsa isfies
S∗
Txν∈A
o ν-almos e e y xwi h B(x, e− ) ⊆B(0, 1).
Theo em 2.21 (Theo em 1.7 o [23]). Le μbe a Radon measu e on Rd. Then o μ-almos
e e y x, e e y angen dis ibu ion a xis a ac al dis ibu ion.
The ollowing is he only p ope y o ac al dis ibu ions ha we equi e di ec ly.
Lemma 2.22. Le Pbe a ac al dis ibu ion. Then o P-a.e. ν, any line Lwi h ν(L) >0
mus con ain he o igin.
P oo . We fi s show ha P-a.e. a omic measu e is he poin mass a he o igin. This is
almos immedia e om he esul s o [23].
Fo a con adic ion, le A ={ν:ν({x}) >0, x =0}and suppose ha P(A) >0.
Le Pbe an e godic componen o Pwi h P(A) >0, and le η∈Abe a uni o mly
scaling measu e gene a ing P. Indeed, by [23, Theo em 1.6], P-almos e e y measu e is
uni o mly scaling. Le xbe such ha η({x}) >0. Since η|{x}η, also η|{x}gene a es
Pby an applica ion o he Lebesgue-Besico i ch diffe en ia ion heo em, whence Pis
suppo ed on he poin mass a he o igin. In pa icula , P(A) =0, a con adic ion.
Now, o p o e he s a emen o he lemma, suppose ha he e exis s a se Bwi h
P(B) >0such ha o e e y ν∈B, he e exis s a line Lνwi h ν(Lν) >0and 0 ∈ Lν.
Le Pbe an e godic componen o Pwi h P(B) >0, and le η∈Bbe a uni o mly
scaling measu e gene a ing P. Now, since η(Lη) >0, also he measu e η|Lηgene a es
P. Le Ldeno e he line Lη ansla ed so ha i con ains he o igin. Clea ly, all angen
measu es o η|Lηa e suppo ed on L, whence Pis suppo ed on measu es which a e
suppo ed on L. Since any line no con aining he o igin in e sec s Lin a mos one
poin , such a line has P-almos su ely ze o measu e by he abo e. Thus P(B) =0
which is a con adic ion.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 23
2.8. Condi ional measu es on lines
Fo a measu e μon R2and θ∈RP1, le μ =´μθ
xdπθμ(x)deno e he disin eg a ion
o μwi h espec o πθ. I is well-known ha o almos e e y x ∈θ, he measu e μθ
xis
suppo ed on he line x +θ⊥, and ha μθ
xis he limi o no malized es ic ions o μon
hinne and hinne ubes cen e ed a x +θ⊥. I will be use ul o us o know ha hese
ubes can be eplaced by p eimages o se s o ela i ely la ge measu e.
Lemma 2.23. Le μbe a measu e on R2, le δ>0and θ∈RP 1. Fo e e y >0, le
I(x, , δ)={I⊆B(x, ): πθμ(I)
πθμ(B(x, )) ≥δ}.
Then o πθμ-almos e e y x, we ha e
lim
→0sup
I∈I(x, ,δ)
dLP μπ−1
θ(I),μ
θ
x=0.
P oo . Fo e e y n ∈N, le En⊆θbe he compac se gi en by Lusin’s heo em wi h
πθμ(En) ≥1 −1/n, on which he unc ion y→ μθ
yis con inuous. Since πθμ(n∈NEn) =
1, i suffices o p o e he s a emen o almos e e y x ∈En, o e e y n ∈N. Now, o
almos e e y x ∈En, i B:= B(x, )and I∈I(x, , δ),
πθμ(I∩En)
πθμ(I)=1−πθμ(I En)
πθμ(I)
≥1−πθμ(B En)
πθμ(B)
πθμ(B)
πθμ(I)
≥1−1
δ
πθμ(B En)
πθμ(B)
=1−o(1)/δ
by an applica ion o he Lebesgue-Besico i ch diffe en ia ion heo em. He e o(1) deno es
a quan i y ha anishes as →0.
The e o e, o any Bo el se A ⊆R2, I∈I(x, , δ)and ε >0,
μπ−1
θI(Aε)= 1
πθμ(I)μ(π−1
θI∩Aε)
=1
πθμ(I)ˆ
I
μθ
y(Aε)dπθμ(y)
≥1−o(1)/δ
πθμ(I∩En)ˆ
I∩En
μθ
y(Aε)dπθμ(y)
≥(1 −o(1)/δ)(μθ
x(A)−ε)
24 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
i is small enough, by con inui y o y→ μθ
y. Simila ly,
μπ−1
θI(A)≤1
πθμ(I∩En)ˆ
I∩Eε
μθ
y(Aε)dπθμ(y)+o(1)/δ ≤μθ
x(Aε)+o(1)/δ +ε,
so dLP(μπ−1I, μθ
x) ≤2ε o small enough . Taking ε →0 comple es he p oo .
3. On local en opy a e ages
In his sec ion, we ecall he local en opy a e ages o [29]and in oduce he diffe en
no ions o magnifica ions o measu es ha we use. Le μand νbe sel -affine measu es
associa ed o i e a ed unc ion sys ems Φ ={ϕi(x) =Aix +ai}i∈Γand Ψ ={ψj(x) =
Bjx +bj}j∈Λ, and deno e by ¯μ, ¯ν he associa ed Be noulli measu es. In his sec ion, we
impose no condi ions on Φand Ψo he han ha Ai <1and Bj <1 o e e y i, j.
Le Nbe a posi i e in ege . Fo e e y i ∈ΓNand j ∈ΛN, we define he “s opping
imes”
ik=ik(i)=min{n∈N:Ai|n≤2−kN },
ik=ik(j)=min{n∈N:Bj|n≤2−kN }.
Al hough he s opping imes on ΓNand ΛNa e deno ed by he same le e ik, he
choice o domain will always be clea om he con ex : o example, i|ik:= i|ik(i)
and j|ik:= j|ik(j). No e also ha we omi he dependence on N om he no a ion: In
p ac ice, Nwill always be a la ge in ege fixed be o ehand. Below, we collec he diffe en
no ions o scale-kmagnifica ions o μand ν. No e ha hey also depend on N.
No a ion 3.1. Le N∈N. Fo each i ∈ΓN, j ∈ΛNand k∈N, le
νj|ik:= S∗
kN TΠ(j)ψj|ikν,
μi|ik:= S∗
kN TΠ(i)ϕi|ikν,
μDkN (Π(i)) := FμDkN (Π(i)),
whe e Fdeno es he unique homo he y aking DkN(Π(i)), he le el-kN dyadic squa e
ha con ains Π(i), on o [−1, 1)2.
No e ha μi|ikand νj|ika e measu es suppo ed on ellipses ha con ain he o igin
and whose majo semi-axes ha e leng h compa able o 1and a e o ien ed in he di ec ions
θ(Ai|ik)and θ(Bj|ik), espec i ely. On he o he hand, μDkN (Π(i)) is a measu e suppo ed
on [−1, 1]2, he “dyadic magnifica ion” o μ. We equi e he ollowing o m o he local
en opy a e ages o [29].
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 31
Le μand νbe ully suppo ed sel -affine measu es associa ed o Φand Ψwi h dim μ >
1 >dim ν. Fo any N0∈N he e exis s an in ege N≥N0such ha o ¯μ-a.e. i ∈ΓN
and ¯ν-a.e. j ∈ΛN, we ha e
lim in
n→∞
1
n
n−1
k=0
1
NHN(π2μDkN (Π(i)) ∗π2νj|ik)≥min{1,dim μ−1+dimν}−ε.
While he domina ion condi ion could be elaxed om Claim 4.3 by sligh ly modi ying
he s a emen , in he p oo o Claim 4.4 i plays a c ucial ole. These claims a e p o ed
in Sec ion 6. We now show how o conclude he p oo o Theo em 1.1.
P oo o Theo em 1.1.Le Φ ={ϕi}i∈Γand Ψ ={ψj}j∈Λbe as in he hypo hesis,
and suppose ha he e exis ully suppo ed sel -affine measu es μand νsuch ha
dim μ ≥dim νand dim(μ ∗ν) <min{2, dim μ +dimν}. By Theo em 1.2 we ha e dim μ >
1 >dim ν. Fo a con adic ion, suppose ha
{|λ1(Ai)|:i∈Γ}∪{|λ2(Bj)|:j∈Λ}
is no an a i hme ic se . I is no difficul o see ha he e mus hen exis a pai
(i, j) ∈Γ ×Λsuch ha
log |λ1(Ai)|
log |λ2(Bj)|∈ Q.
Le ε >0be small, and conjuga e Φand Ψ h ough a linea map as in Claims 4.3 and
4.4. Since Φand Ψa e conjuga ed h ough he same map, i is easy o see ha a pai
(i, j) ∈Γ ×Λand measu es μand νas abo e also exis o he conjuga e IFSs. Now, by
[29, Lemma 4.3], we ha e
dim μ= lim in
n→∞
1
n
n−1
k=0
1
NHN(μDkN (Π(i)))
o ¯μ-almos e e y i ∈ΓNand any N∈N, so by (4.4)and Claims 4.3 and 4.4, o any
N0∈N he e exis s N≥N0such ha
lim in
n→∞
1
n
n−1
k=0
1
NHN(μDkN (Π(i)) ∗νj|ik)
≥min{1,dim μ−1+dimν}+dimμ−(dim μ−1) −2ε−O(1/N )
=min{2,dim μ+dimν}−3ε
o ¯μ-almos e e y i ∈ΓNand ¯ν-almos e e y j ∈ΛN. I now ollows om Theo em 3.2
ha dim(μ ∗ν) ≥min{2, dim μ +dimν} −4ε, which is a con adic ion i εis small
enough.
32 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
5. The scene y o he sel -affine measu e
In his sec ion, ou aim is o p o e P oposi ion 4.1. We assume h oughou he sec ion
ha μis a sel -affine measu e associa ed o an i e a ed unc ion sys em Φ ={ϕi(x) =
Aix +ai}i∈Γsa is ying he i educibili y, domina ion and s ong sepa a ion condi ions,
and ha dim μ >1. The a gumen s o his sec ion we e inspi ed by he wo k o Kemp on
[32]on a simila esul o a mo e special class o sel -affine measu es, and some o ou
lemmas a e analogous o hose in [32].
5.1. Res ic ions o μon hin ec angles
Le us begin wi h he heu is ics o why magnifica ions o μa e ela ed o slices o μ.
Le i ∈ΓN, le Bbe a small ball cen e ed a Π(i)and le nbe he la ges in ege so
ha ϕi|n(B(0, 1)) ⊇B. By he s ong sepa a ion condi ion, we ha e μB=ϕi|nϕ−1
i|nμB=
ϕi|nμϕ−1
i|nB. The e o e, in o de o us o unde s and he magnifica ions μB, i suffices o
unde s and he measu es μϕ−1
i|nB, he es ic ions o μon he ellipses ϕ−1
i|nBwhose majo
semi-axes ha e leng h compa able o 1and a e pa allel o θ(A−1
i|n). Since Lemma 2.1
asse s ha he ellipses ϕ−1
i|nBge hinne and hinne as ninc eases, he measu e μϕ−1
i|nB
should be close o a slice o μin he di ec ion θ(A−1
i|n), when nis la ge.
In a mo e igo ous app oach, i is be e o wo k wi h ec angles ins ead o ellipses.
Fo i ∈ΓN, θ∈RP1, 2≥ 1≥0, w i e Yi,θ, 1, 2 o he ec angle cen e ed a Π(i)wi h
sideleng hs 2− 1≥2− 2and he longe side o ien ed in he di ec ion θ. Fo a ec angle
Ywi h cen e a xand majo side o ien ed in he di ec ion θ, w i e HY o he map
which ansla es x o he o igin and s e ches TxYon o R−1
θ[−1, 1]2. I ollows om
Lemma 2.23 ha he measu es HYi,θ, 1, 2μYi,θ, 1, 2ha e a fibe s uc u e in he ollowing
sense.
Lemma 5.1. Fo ¯μ×μF-a.e. (i, θ) ∈ΓN×RP 1and any ε, δ, >0, he e exis s >0
such ha i 0 ≤ 1≤ , 2≥ , Qis a ansla e o [0, 1/2]2 ha con ains he o igin and
RθHYi,θ, 1, 2μYi,θ, 1, 2(Q)≥δ,
hen
dLP π2((RθHYi,θ, 1, 2μYi,θ, 1, 2)Q),(μi,θ, 1)π2Q<ε.
Fo he p oo , we eco d he ollowing elemen a y obse a ions.
Lemma 5.2 (Lemma 3.3 o [5]). Fo any , δ>0, he ollowing holds o all small enough
ε ≥ε>0:
I μand νa e p obabili y measu es on [−1, 1]dwi h dLP(μ, ν) <ε
and B=B(x, )
is a closed ball wi h min{μ(B), ν(B)} ≥δand ν(B(x, +ε)) ≤ν(B(x, −ε)) +ε, hen
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 33
dLP(μB,ν
B)<O(ε/δ).
Lemma 5.3. Le μbe a p obabili y measu e on R. I I⊆Ris a closed in e al which
con ains 0, hen μ(x +I) >0 o μ-almos e e y x ∈R.
P oo . The dis ance be ween any wo poin s in E:= {x ∈R :μ(x +I) =0} ∩sp μis
a leas |I|/2 >0. In pa icula , Eis coun able, and so μ(E) ≤x∈Eμ(x +I) =0.
P oo o Lemma 5.1.Recall om (4.1) ha μi,θ =RθTΠ(i)μθ
πθΠ(i), whe e μθ
πθΠ(i)is a
condi ional measu e o μ om he disin eg a ion μ =´μθ
ydπθμ(y) =´μθ
πθΠ(i)d¯μ(i),
and ha μi,θ, =S∗
μi,θ. We claim ha
in {μi,θ, 1({0}×[a, a +1/2]) : −1/2≤a≤0,0≤ 1≤ }>0 (5.1)
o e e y θ∈RP1and ¯μ-almos e e y i ∈ΓN. Indeed, o e e y θ∈RP1and πθμ-almos
e e y y∈θ, he measu e μθ
yexis s and is suppo ed on he line y+θ⊥. I ollows om
Lemma 5.3, applied wi h y+θ⊥in place o Rand o he in e al R−1
θ({0} ×[0, 2− /4]) ⊆
θ⊥, ha
μθ
y(x+R−1
θ({0}×[0,2− /4])) >0
o μθ
y-almos e e y x ∈y+θ⊥. Since πθ(x) =y o e e y x ∈y+θ⊥, i ollows ha
μΠ−1(x),θ({0} ×[0, 2− /4]) =μθ
y(x +R−1
θ({0} ×[0, 2− /4])) >0 o πθμ-almos e e y y
and μθ
y-almos e e y x. Bu since μ =´μθ
ydπθμ(y), his means ha
μi,θ, 1({0}×[0,1/4]) ≥μi,θ({0}×[0,2− /4]) >0
o ¯μ-almos e e y i ∈ΓNand e e y 0 ≤ 1≤ . Analogously, μi,θ, 1({0} ×[−1/4, 0]) >0
o ¯μ-almos e e y iand e e y 0 ≤ 1≤ . Since he in e als o he o m {0} ×[a, a +1/2]
conside ed in (5.1)always con ain ei he {0} ×[−1/4, 0] o {0} ×[0, 1/4], (5.1) ollows.
Thus, by (5.1), o a gi en ε >0, he e exis s c >0and a se Aεwi h ¯μ×μF(Aε) >1 −ε
such ha μi,θ, 1({0} ×[a, a +1/2]) ≥c o e e y (i, θ) ∈Aε, −1/2 ≤a ≤0and 0 ≤ 1≤ .
Since ¯μ×μF(n∈NA1/n) =1, i suffices o p o e he s a emen o (i, θ) ∈Aε, o a
gi en ε >0.
Recall ha π1deno es he o hogonal p ojec ion o he x-axis. Lemma 2.23 ansla ed
o he language o his sec ion s a es ha i ≥0is la ge enough, hen o e e y 2≥ ,
dLP π2(RθHYi,θ, 1, 2μYi,θ, 1, 2)(π1)−1π1(Q),μ
i,θ, 1<ε.
See Fig. 1. Since dim μ >1, μi,θ, 1is non-a omic o ¯μ×μF-almos e e y (i, θ)by Theo-
em 2.7. Mo eo e , (π1)−1π1(Q) ∩(π2)−1π2(Q) =Qand μi,θ, 1(π2Q) ≥c, so Lemma 5.2
combined wi h he abo e asse s ha o a possibly e en la ge ≥0, o e e y 2≥
and e e y 0 ≤ 1≤ ,
34 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
dLP π2(RθHYi,θ, 1, 2μYi,θ, 1, 2)Q,(μi,θ, 1)π2Q<ε
which is wha we wan ed o p o e.
θ
←− π2
π2Q
(0,0)
Q
HYi,θ, 1, 2Yi,θ, 1, 2
(π1)−1π1(Q)
Fig. 1. The measu e HYi,θ, 1, 2μYi,θ, 1, 2p ojec s close o a slice o μunde π2, e en when es ic ed on o
(π1)−1π1(Q).
We will nex ela e he measu es μDkN (Π(i)) o he measu es HYi,θ, 1, 2μYi,θ, 1, 2. To
make s a emen s mo e economic, we in oduce some addi ional no a ion.
5.2. Magnifica ions o μ
Fo a ∈Γ∗, le Aa=UaDaV−1
adeno e he singula alue decomposi ion, whe e Ua, Va
a e o hogonal and Da=diag(α1(Aa), α2(Aa)). Recall he defini ion o he sequence
(k)k∈N om (4.2). Th oughou he ollowing, we will use he sho -hand no a ion
Qi,θ,k := Yσki,θ,kN−2+log α1(i|k),kN−2+log α2(i|k).
Then he se Qi,θ(A−1
i|k
),k ⊂R2is he smalles ec angle ha con ains he ellipse
ϕ−1
i|k
(B(Π(i), 2−kN+2)).
P oposi ion 5.4. Fo e e y i ∈ΓN, e e y la ge enough N∈Nand e e y θ∈
B(θ(i)⊥, 1/5), he e exis s a sequence o non-singula linea maps (Li,θ,k)k∈Nsuch ha
S∗
kN−1TΠ(i)μ=S∗
1Ui|kV−1
i|k
Li,θ,kHQi,A−1
i|k
θ,k μQi,A−1
i|k
θ,k
o all la ge enough k.
An impo an poin o he p oposi ion is ha he di ec ion θcan be chosen a bi a ily
om a la ge se . In p o ing his we equi e he ollowing geome ic lemma, analogous o
[32, Lemma 8.2]. W i e Ei,θ,k ⊆Qi,θ,k o he la ges ellipse con ained in Qi,θ,k. No e
ha Ei,θ(A−1
i|k
),k =ϕ−1
i|k
(B(Π(i), 2−kN+2)).
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 35
Lemma 5.5 (Fig. 2). Fo ¯μ-almos e e y i ∈ΓNand e e y θ∈B(θ(i)⊥, 1/5), o e e y
la ge enough k,
S−1TΠ(σki)Ei,θ(A−1
i|k
),k ⊆TΠ(σki)Qi,A−1
i|k
θ,k.
Fig. 2. The ellipse Ei,θ(A−1
i|k
),k (ligh g ay) fi s comple ely inside he ec angle Qi,A−1
i|k
θ,k a e he axes a e
scaled by 1/2.
P oo . Fo any θ∈RP1, i ∈ΓNand k∈N, we ha e
an d(A−1
i|kθ, θ(A−1
i|k)) = α1(i|k)
α2(i|k) an d(θ, θ(Ai|k)⊥).(5.2)
See Fig. 3.
1A−1
i|k→
α2(i|k)−1
α1(i|k)−1
ββ
Fig. 3. He e β=d(θ, θ(Ai|k)⊥)which is aken o β=d(A−1
i|kθ, θ(A−1
i|k)) by A−1
i|k. I he side o leng h 1
is pa allel o θ(Ai|k)⊥(which is he majo expanding di ec ion o A−1
i|k), hen i s image h ough A−1
i|kis o
leng h α1(i|k)−1.
Le η=d(A−1
i|kθ, θ(A−1
i|k)) and define X=[−α1(i|k), α1(i|k)] ×[−α2(i|k), α2(i|k)]
and Y=R−1
η([−α1(i|k)
2,
α1(i|k)
2] ×[−α2(i|k)
2,
α2(i|k)
2]). I is no difficul o see ha Y⊆X
i we ha e sin η·α2(i|k)
2+α1(i|k)
2≤α1(i|k), o equi alen ly,
sin η≤α1(i|k)
α2(i|k).
Since η→0as k→∞by Lemma 2.3, we ha e sin η≤2 an η o la ge enough k.
The e o e, o ¯μ-almos e e y i ∈ΓNand e e y θ∈B(θ(i)⊥, 1/5), i ollows om (5.2)
and Lemma 2.2 ha
sin η≤2α1(i|k)
α2(i|k) an d(θ, θ(Ai|k)⊥)≤α1(i|k)
α2(i|k)2 an(1/4) <α1(i|k)
α2(i|k),
36 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
o all la ge enough k.
P oo o P oposi ion 5.4.Le Nbe a la ge enough in ege so ha
2−N<min{d(ϕi(B(0,1)),ϕ
j(B(0,1)) : i, j ∈Γ},
le (i, θ) ∈ΓN×RP1, le kbe la ge, and le kbe defined as in (4.2). W i e
Bi|k:= Ui|kD−1
i|k
U−1
i|k
. Recalling ha Ei,θ(A−1
i|k
),k =(ϕi|k)−1B(Π(i), 2−kN+2), i is
no difficul o see ha SkN−2B−1
i|k
is he linea map which scales he o igocen ic ellipse
Ui|kV−1
i|k
Tϕ−1
i|k
Π(i)Ei,θ(A−1
i|k
),k on o B(0, 1) wi hou o a ing i .
By he s ong sepa a ion condi ion, wi h his no a ion we may w i e
S∗
kN−2TΠ(i)μ=S∗
kN−2TΠ(i)ϕi|kμEi,θ(A−1
i|k
),k
=S∗
kN−2B−1
i|k
Bi|kAi|kTϕ−1
i|k
Π(i)μEi,θ(A−1
i|k
),k
=S∗
kN−2B−1
i|k
Ui|kV−1
i|k
Tϕ−1
i|k
Π(i)μEi,θ(A−1
i|k
),k
=Ui|kV−1
i|k
HQi,θ(A−1
i|k
),k μEi,θ(A−1
i|k
),k ,(5.3)
by swi ching he o de o scaling and o a ion in he las equali y. This is almos
he ep esen a ion we a e seeking. The eason we a e no ye con en wi h his
is ha we would ul ima ely like o apply Lemma 5.1 o say ha he measu e
Rθ(A−1
i|k
)HQi,θ(A−1
i|k
),k μEi,θ(A−1
i|k
),k is close o μσki,θ(A−1
i|k
), o some ≥0. Howe e ,
we a e able o say his only i θ(A−1
i|k
)is eplaced by some θwhich is d awn andomly
wi h espec o μF. Al hough he sequence (θ(A−1
i|k
))k∈Ndoes in ac equidis ibu e
o μF, o ¯μ-almos e e y i ∈ΓN, i is only in he weak-∗sense and he lack o any
con inui y o he unc ion (i, θ) → μi,θ makes i difficul o say any hing abou he ela-
ionship o Rθ(A−1
i|k
)HQi,θ(A−1
i|k
),k μEi,θ(A−1
i|k
),k and μσki,θ(A−1
i|k
). The e o e, we de o e he
es o he p oo o he echnical wo k o eplacing he measu e HQi,θ(A−1
i|k
),k μEi,θ(A−1
i|k
),k
by HQi,A−1
i|k
θ,k μQi,A−1
i|k
θ,k in (5.3), whe e θcan be d awn eely om a se o posi i e
μF-measu e, so ha we may e en ually apply Lemma 5.1.
I we now le θ∈B(θ(i)⊥, 1/5), hen by Lemma 5.5 we ha e
S−1TΠ(σki)Ei,θ(A−1
i|k
),k ⊆TΠ(σki)Qi,A−1
i|k
θ,k (5.4)
o e e y la ge enough k. Using he gene al ac ha μX=(μY)Xwhene e X⊆Y⊆R2,
we ob ain om (5.4) ha
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 37
TΠ(σki)μEi,θ(A−1
i|k
),k S−1TΠ(σki)Ei,θ(A−1
i|k
),k
=TΠ(σki)μQi,A−1
i|k
θ,k S−1TΠ(σki)Ei,θ(A−1
i|k
),k
.
Pushing bo h measu es o wa d unde HQi,θ(A−1
i|k
),k ◦T−1
Π(σki)and using he iden i y
B(0, 1) =HQi,θ(A−1
i|k
),k T−1
Π(σki)TΠ(σki)Ei,θ(A−1
i|k
),k which is immedia e om he defini-
ions o Hand E, we ob ain
HQi,θ(A−1
i|k
),k μEi,θ(A−1
i|k
),k B(0,1/2)
=HQi,θ(A−1
i|k
),k μQi,A−1
i|k
θ,k B(0,1/2)
.(5.5)
See Fig. 4.
Qi,A−1
i|k
θ,k
Ei,θ(A−1
i|k
),k
H... →
Fig. 4. The p ecise “loca ion” o he line A−1
i|k
θin RP 1is easie o con ol han ha o θ(A−1
i|k
)e en hough
he lines a e e y close o each o he . Because o his, we wan o conside es ic ions o μon Qi,A−1
i|k
θ,k
ins ead o on Ei,θ(A−1
i|k
),k.
W i ing
Li,θ,k := HQi,θ(A−1
i|k
),k H−1
Qi,A−1
i|k
θ,k
and combining (5.5)wi h (5.3), we ob ain
S∗
kN−1TΠ(i)μ=S∗
1Ui|kV−1
i|k
Li,θ,kHQi,A−1
i|k
θ,k μQi,A−1
i|k
θ,k
o all la ge enough k. This comple es he p oo .
No e ha because DkN (Π(i)) ⊆B(Π(i), 2−kN+1), i ollows ha
μDkN (Π(i)) =(S∗
kN−1TΠ(i)μ)Q=(S∗
1Ui|kV−1
i|k
Li,θ,kHQi,A−1
i|k
θ,k μQi,A−1
i|k
θ,k )Q(5.6)
38 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
whe e Qis a ansla e o [0, 1/2]2 ha con ains he o igin. He e and in he ollowing,
i νis a measu e on he plane and Qis a squa e, we w i e νQ=FνQwhe e Fis he
unique homo he y sending he closu e o Qon o [−1, 1]2. I he map Ui|kV−1
i|k
Li,θ,k
we e jus he o a ion RA−1
i|k
θ, hen all ha would be le o ob ain he fibe s uc u e
o μDkN (Π(i)) we e o combine (5.6)and Lemma 5.1; See Fig. 5.
S1Ui|kV−1
i|k
Li,θ,kHQi,A−1
i|k
θ,k Qi,A−1
i|k
θ,k
Q
(0,0)
Fig. 5. Fo us o be able o di ec ly apply Lemma 5.1, he abo e pa allelog am would ha e o be a squa e
simila o Q.
Un o una ely, because o he in ol emen o he map Li,θ,k, he map Ui|kV−1
i|k
Li,θ,k
is no me ely a o a ion, and some echnical wo k is equi ed o deal wi h his addi ional
dis o ion. We begin by b eaking he map Ui|nV−1
i|nin o h ee componen s: The eflec ion,
and wo o a ions which do no eflec .
5.3. The dis o ion Ui|kV−1
i|k
Li,θ,k
Fo a linea map A :R2→R2and θ∈RP1, le A|θ:θ→R2deno e he es ic ion
o Aon o θ. Fo e e y θ∈RP 1, le eθdeno e he uni ec o o θwi h non-nega i e y-
coo dina e. Le O⊂ΓN×RP1be he open se o hose (i, θ) o which eA−1
i0θ, A−1
i0eθ <
0. He e ·, · deno es he Euclidean inne p oduc . Recall ha Mdeno es he map
(i, θ) → (σi, A−1
i0θ). Define he unc ion ρ :N×(Γ ×RP 1) →{−1, 1},
ρ(n, (i,θ)) =
n
k=1
(−1)1O(Mk(i,θ)) (5.7)
whe e 1Odeno es he indica o o O. The map ρcap u es he eflec ions done by A−1
i|n
on he line θ, o by Ai|non he line A−1
i|nθ. Indeed, o any x ∈θ, we may decompose
A−1
i|nas
A−1
i|nx=A−1
i|nxR−1
A−1
i|nθρ(n, (i,θ))Rθx. (5.8)
In o he wo ds, fi s o a e x o he y-axis, apply he possible eflec ions, o a e he y-
axis on o he line A−1
i|nθ, and finally scale. The map ρis easily seen o sa is y he cocycle
equa ion
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 39
ρ(n+k,(i,θ)) = ρ(n, Mk(i,θ))ρ(k,(i,θ))
o e e y n, k∈N.
Lemma 5.6. Le i ∈ΓNand le Ui|nDi|nV−1
i|nbe he singula alue decomposi ion o Ai|n.
Fo any θ∈RP 1 {θ(i)}, we ha e
lim
n→∞ Ui|nV−1
i|n|A−1
i|nθ−R−1
θ(i)⊥ρ(n, (i,θ))RA−1
i|nθ|A−1
i|nθ=0.
P oo . Le Bi|n:= Ui|nD−1
i|nU−1
i|nand no e ha
Ai|n=B−1
i|nBi|nAi|n=B−1
i|nUi|nV−1
i|n.
In pa icula , Ui|nV−1
i|n=Bi|nAi|n.
Now, by (5.8),
Ai|n|A−1
i|nθ=Ai|n|A−1
i|nθR−1
θρ(n, (i,θ))RA−1
i|nθ|A−1
i|nθ.(5.9)
On he o he hand, i ollows om Lemma 2.5 and some basic geome y ha
Bi|n|θ−(−1)knA−1
i|n|θR−1
θ(i)⊥Rθ|θ
→0
as n →∞, o some sequence (kn)n∈{1, 2}N. Since Bi|nis posi i e defini e and θ=θ(i),
he sequence (−1)knhas o be e en ually cons an . By abso bing he e en ual alue o
his sequence o he defini ion o ρ(n, (i, θ)), we may wi hou loss o gene ali y assume
ha
lim
n→∞ A−1
i|n|θ−1Bi|n|θ=R−1
θ(i)⊥Rθ|θ.(5.10)
Thus, combining (5.9)and (5.10), we ha e
lim
n→∞ Bi|nAi|n|A−1
i|nθ−R−1
θ(i)⊥ρ(n, (i,θ))RA−1
i|nθ|A−1
i|nθ=0
which is wha we wan ed o show.
I emains o s udy he beha io o linea map Li,θ,k om he s a emen o P opo-
si ion 5.4, as k→∞. The con en o he ollowing lemma is ha Ui|kV−1
i|k
Li,θ,k akes
he squa e HQi,A−1
i|k
θ,k Qi,A−1
i|k
θ,k =R−1
A−1
i|k
θ[−1, 1]2on o a pa allelog am o bounded ec-
cen ici y and one side in di ec ion θ(i).
Lemma 5.7. Fo e e y i ∈ΓNand θ∈RP1 {θ(i)},
Ui|kV−1
i|k
Li,θ,k −R−1
θ(i)⊥ρ(k,(i,θ))Fk
θ,iRA−1
i|k
θ→0
40 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
as k→∞, whe e Fk
θ,i=akbk· an d(θ, θ(i)⊥)
01
o some ak, bk∈{−1, 1}.
P oo . Recall ha Li,θ,k =HQi,θ(A−1
i|k
),k H−1
Qi,A−1
i|k
θ,k
. As he map Li,θ,k ac s on he
se R−1
A−1
i|k
θ[−1, 1]2, i is fi s aken o he hin ec angle H−1
Qi,A−1
i|k
θ,k
R−1
A−1
i|k
θ[−1, 1]2(see
Fig. 6), and hen s e ched on o a pa allelog am by HQi,θ(A−1
i|k
),k .
HQi,θ(A−1
i|k
),k →
Fig. 6. The se H−1
Qi,θ(A−1
i|k
),k
R−1
θ(A−1
i|k
)[−1, 1]2is depic ed as he hin whi e ec angle, while he se
H−1
Qi,A−1
i|k
θ,k
R−1
A−1
i|k
θ[−1, 1]2is depic ed as he hin g ay ec angle.
The angle be ween he wo hin ec angles in Fig. 6is d(θ(A−1
i|k
), A−1
i|k
θ), so
by (5.2)and basic geome y, he Hausdo ff dis ance be ween he pa allelog ams
Li,θ,kR−1
A−1
i|k
θ[−1, 1]2and R−1
θ(A−1
i|k
)Fk
θ,i[−1, 1]2 ends o 0as k→∞, whe e in he defi-
ni ion o Fk
θ,iwe ha e ak=1and bk∈{−1, 1}depends on he o de o θ(A−1
i|k
)and
A−1
i|k
θ. In pa icula ,
Li,θ,k −R−1
θ(A−1
i|k
)Fk
θ,iRA−1
i|k
θ→0
as k→∞. I ollows om Lemma 2.3 ha d(θ(A−1
i|k
), A−1
i|k
θ) →0, so using Lemma 5.6
and inco po a ing he possible eflec ion o he line θ(i)as he alue o akin he
defini ion o Fk
θ,icomple es he p oo .
5.4. P oo o P oposi ion 4.1
We will now explain how o combine P oposi ion 5.4 and Lemmas 5.1 and 5.7 o p o e
P oposi ion 4.1 and ob ain he fibe s uc u e o μDkN (Π(i)). In (5.6)we no ed ha by
P oposi ion 5.4, he dyadic magnifica ions o μha e he o m
μDkN (Π(i)) =(S∗
kN−1TΠ(i)μ)Q=(S∗
1Ui|kV−1
i|k
Li,θ,kHQi,A−1
i|k
θ,k μQi,A−1
i|k
θ,k )Q
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 47
P oo . Since Z
Φis conjuga e o Zh
Φand conjuga e flows ha e he same eigen alues, i
suffices o p o e he s a emen o Zh
Φ. Le β=0be an eigen alue o Zh
Φ. By Lemma 2.13,
he e exis s n ∈Zsuch ha nβ is an eigen alue o (Zh
Φ)+.
Le i ∈Γ, and le i0=...iii ...∈ΓZ. Now, |λ1(Ai)|−1=|λ2(A−1
i)| =A−1
i|θ(i−
0) =
Ai|θ−(i−
0)−1. In pa icula ,
(i0)=−log Ai|A−1
iθ−(i−
0)=−log |λ1(Ai)|.
F om (6.2)i is clea ha also h(i0) =− log |λ1(Ai)|.
Since nβ is an eigen alue o (Zh
Φ)+, P oposi ion 2.12 asse s ha he e exis s a con-
inuous eigen unc ion φ o nβ, so we ha e
φ(Ts(i, )) = e(nβs)φ(i, ) (6.3)
o e e y (i, ) ∈(Zh
Φ)+and s ≥0. Since (Zh
Φ)+is e godic, |φ|is cons an e e ywhe e, so
we may le ψ:(Zh
Φ)+→Rbe he eal- alued unc ion defined by φ(i, ) =|φ|e(ψ(i, ))
o e e y (i, ) ∈(Zh
Φ)+. We ob ain om (6.3) ha
ψ(Ts(i, )) = nβs +ψ(i, )+m(i, )
o some in ege - alued unc ion m :(Zh
Φ)+→Z. Inse ing he alue s =h(i), we ob ain
ψ(σi, )=nβh(i)+ψ(i, )+m(i, )
which is equi alen o
h(i)=(nβ)−1m(i, )+(nβ)−1ψ(i, )−(nβ)−1ψ(σi, ).
In pa icula , h(i0) =(nβ)−1m(i0, ) ∈β−1Qsince σ(i0) =i0. Howe e , we saw abo e
ha h(i0) =− log |λ1(Ai)|, whence i ollows ha β∈(log |λ1(Ai)|)−1Q.
Lemma 6.3. I Ψsa isfies he domina ion condi ion, hen he eigen alues o Z
Ψa e
con ained in he se
j∈Λ
(log |λ2(Bj)|)−1Q
P oo . Le j0=...jjj.... Then j0is a fixed poin o σ, and
g(j0)=−log B∗
j|θ∗(j−
0)=−log |λ2(B∗
j)|=−log |λ2(Bj)|
and by eplacing Z
Ψwi h a conjuga e flow, we can p oceed exac ly as in he p oo o
he p e ious claim.
48 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
P oo o P oposi ion 6.1.By Lemmas 6.2 and 6.3 and he assump ion o P oposi ion 6.1,
he flows Z
Φand Z
Ψha e no common eigen alues o he han 0. Thus by P oposi ion 2.10,
he p oduc Z
Φ×Z
Ψis e godic. Since i has a mos coun ably many eigen alues by
Lemma 2.9, o any N0∈N he e exis s a eal numbe N>N
0such ha i is also
e godic unde he disc e e- ime map TN. By a change o coo dina es, we may suppose
u he ha Nis an in ege .
Now, since we ha e he domina ion assump ion in place, he sys em Z
Φ×Z
Ψis a
ac o o he sys em Z
Φ×Z
Ψ h ough he map
(i, ,j,s)→ (i+,θ−(i−), ,j+,θ∗(j−),s).
Since ac o maps p ese e e godici y, his p o es he s a emen .
6.2. Dynamics o he sequence (μDkN (Π(i)))k∈N
Le Φ ={ϕi(x) =Aix +ai}i∈Γbe an affine i e a ed unc ion sys em sa is ying
he i educibili y, domina ion and s ong sepa a ion condi ions, le μbe a sel -affine
measu e associa ed o Φand le ZΦbe as in he p e ious sec ion. The eason o e-
qui ing domina ion and s ong sepa a ion in his sec ion (e en hough hey we e no
equi ed in he cons uc ion o ZΦ) is ha hey allow us o use P oposi ion 4.1 o ela e
(π2μDkN (Π(i)))k∈N o an o bi o a poin in ZΦ.
Lemma 6.4. Fo ¯μ-almos e e y i ∈ΓNand e e y θ=θ(i), he e exis s a cons an
C(i, θ) >0such ha
lim
n→∞
Ai|n|A−1
i|nθ
α1(Ai|n)=2
C(i,θ).
P oo . Since Ai|n|A−1
i|nθ =|Ai|nB(0, 1) ∩θ|, we ha e
Ai|n|A−1
i|nθ
α1(Ai|n)=|Ai|nB(0,1) ∩θ|
|Ai|nB(0,1) ∩θ(Ai|n)⊥|
=|Ai|nB(0,1) ∩θ|
|Ai|nB(0,1) ∩θ(i)⊥|
|Ai|nB(0,1) ∩θ(i)⊥|
|Ai|nB(0,1) ∩θ(Ai|n)⊥|.
He e, o ¯μ-almos e e y i, we ha e limn→∞ |Ai|nB(0,1)∩θ(i)⊥|
|Ai|nB(0,1)∩θ(Ai|n)⊥|=1by Lemma 2.2 and
limn→∞ |Ai|nB(0,1)∩θ|
|Ai|nB(0,1)∩θ(i)⊥|=(cosd(θ, θ(i)⊥))−1by Lemma 2.2 and basic geome y. This
comple es he p oo wi h C(i, θ) =− log(cos d(θ, θ(i)⊥)).
In pa icula , o ¯μ×μF-almos e e y (i, θ) ∈ΓN×RP1,
dLP(μMk(i,θ),kN+log α1(i|k),μ
Mk(i,θ),kN+log Ai|k|A−1
i|k
θ−C(i,θ))→0 (6.4)
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 49
as k→∞, whe e he sequence (k)k∈N=(k(i))k∈Nis as in (4.2). No e ha o la ge
enough k,
k=max{n:α1(Ai|n)≥2−(k−1)N}
=max{n:Ai|n|A−1
i|nθ−1≤2(k−1)N−C(i,θ)}
=max{n:−log Ai|n|A−1
i|nθ≤(k−1)N−C(i,θ)}.
Combining his wi h he elemen a y obse a ion ha
Ai1Ai2|A−1
i2A−1
i1θ=Ai1|A−1
i1θ·Ai2|A−1
i2A−1
i1θ
we see ha in ZΦwe ha e he iden i y
(i,θ,u,(k−1)N−C(i,θ))
=(Mk(i,θ),ρ(k,(i,θ))u, (k−1)N−C(i,θ)+logAik|A−1
i|k
θ) (6.5)
o e e y (i, θ, u) ∈ΓZ×RP1×{−1, 1}.
Le Fdeno e he map ZΦ→P(R2),
(i,θ,u, )→ 4uμi,θ, +N−1,(6.6)
o e e y (i, θ, u, )wi h 0 ≤ < (i, θ), whe e 4μ(·) := μ(4−1·). In sho , we will
commen on he eason why we scale by 4.
By (6.5), we ha e
F(i,θ,Id,(k−1)N−C(i,θ))
=4ρ(k,(i,θ))μMk(i,θ),kN−1+C(i,θ)+log Aik|A−1
i|k
θ.(6.7)
Le ε >0. I we now eplace Φby a linea ly conjuga e sys em as in P oposi ion 4.1 and
μis any ully suppo ed sel -affine measu e associa ed o Φ, hen by equa ions (6.4)and
(6.7), he ollowing holds:
Fo ¯μ-a.e. i ∈ΓNand all θin a se o posi i e μF-measu e, he e exis s a sequence o
in e als (Ik)kwi h |Ik| =2and a se Nε⊆Nsuch ha lim in n→∞ #(Nε∩[0,n])
n≥1 −ε
and
dLP(π2μDkN (Π(i)),(F(i,θ,1,(k−1)N−C(i,θ)))Ik)<ε (6.8)
o e e y k∈N
ε. No e ha because |Ik| =2, he ope a ions (·)Ikand (·)Ikdiffe only by
a ansla ion. He e i is con enien ha we in ol ed he cons an 4in (6.6): O he wise,
he in e als Ikwould be o leng h 1/2and he ope a ion (·)Ikwould in ol e scaling in
addi ion o ansla ion, which would cause mino echnical incon eniences in he u u e.
50 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
6.3. Dynamics o he sequence (νj|ik)k∈N
Le Ψ ={ψj(x) =Bjx +bj}j∈Λbe an affine i e a ed unc ion sys em sa is ying he
o al i educibili y and hype bolici y condi ions. We ema k ha in his subsec ion, no
domina ion o any sepa a ion condi ion is equi ed. Simila ly as in he p e ious sec ion,
ou aim is o ela e he sequence (νj|ik)k∈N o an o bi o a poin in ZΨ. In ac , i
u ns ou o be mo e use ul o s udy he sequence (πθνj|ik)k∈N, o e e y θ∈RP1. As
we will demons a e in his sec ion, his sequence is ela ed o ZΦ h ough he unc ion
G :ZΨ→P(R2),
(j,θ, , )→ S∗
RθπθTΠ(j)ν. (6.9)
Fo any θ∈RP1, x ∈R2and ma ix B, i ollows om he iden i y Bx, eθ =x, B∗eθ
ha
πθ(Bx)=±R−1
θB∗|θRB∗θπB∗θ(x),(6.10)
we e he sign is nega i e i and only i eB∗θ, B∗eθ <0, ecalling ha eθdeno es he
uni ec o o θwi h non-nega i e y-coo dina e. In pa icula ,
|πθ(Bj|kB(0,1))|=B∗
j|k|θ=B∗
j0|θ·...·B∗
jk|B∗
jk−1...B∗
j0θ(6.11)
and since TΠ(j)◦ϕj|k=Bj|k◦TΠ(σkj), we ha e
πθνj|k=πθS∗
kN TΠ(j)ψj|kν
=R−1
θρ(k,(j,θ))S∗
kN+log B∗
j|k|θRB∗
j|kθπB∗
j|kθTΠ(σkj)ν, (6.12)
whe e ρdeno es he cocycle ha cap u es he sign o he igh -hand side o (6.10)and
was defined in (6.1). F om (6.11)and Lemma 2.2 i eadily ollows ha
lim
n→∞
Bj|n
B∗
j|n|θ=cosd(θ, θ(j)) (6.13)
o ¯ν-almos e e y j ∈ΛNand θ=θ(j)⊥. In pa icula , ecalling ha
ik=ik(j)=min{n∈N:Bj|n≤2−kN }
=min{n∈N:B∗
j|n|θ≤2−kN+log cos(d(θ,θ(j)))},
o e e y la ge enough k, i ollows om (6.12)and (6.13) ha
lim
k→∞ dLP(πθνj|ik,R
−1
θG(T(j,θ,1,kN −log cos d(θ, θ(j))))) = 0 (6.14)
o ¯ν-almos e e y j ∈ΛNand e e y θ=θ(j)⊥.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 51
6.4. P oo o Claims 4.2, 4.3 and 4.4
We now p oceed o p o e Claims 4.2, 4.3 and 4.4 using (6.8), (6.14)and P opo-
si ion 6.1. Recall ha (6.8)holds when Φsa isfies he i educibili y, domina ion and
s ong sepa a ion condi ions, while (6.14)holds when Ψsa isfies he o al i educibili y
and hype bolici y condi ions.
P oo o Claim 4.2.Le μand νbe as in he s a emen o he claim. We s ess ha in
his p oo , only o al i educibili y, hype bolici y and exponen ial sepa a ion is assumed
o bo h measu es. I will be con enien o eplace he unc ion η→ HN(η)by he weak-∗
con inuous subs i u e
˜
HN:η→ ´1
0HN(δx∗η) dx. I ollows om Lemma 2.16 ha
|˜
HN(η) −HN(η)| ≤O(1), whence any bounds o
1
N˜
HNhold also o
1
NHN, up o an
addi i e e o o O(1/N ).
Recall ha μi|ik=S∗
kN TΠ(i)ϕi|ikμand νj|ik=S∗
kN TΠ(j)ψj|ikν. By Lemma 2.1, any
accumula ion poin s o SkN TΠ(i)ϕi|ikand SkN TΠ(j)ψj|ikas k→∞a e affine maps o
ank 1, wi h anges θ(i)and θ(j), espec i ely. In pa icula , o ¯μׯν-almos e e y
(i, j),
lim
k→∞ dLP(μi|ik∗νj|ik,π
θ(i)μi|ik∗πθ(j)νj|ik) = 0 (6.15)
Recall he defini ion o he unc ion G om (6.9), and no e ha by Theo em 2.6,
dim G(j, θ, , ) =min{1, dim ν} o λΨ-almos e e y (j, θ, , ). This is he only pa in
he p oo whe e he exponen ial sepa a ion is used. Since he flows ZΦand ZΨha e
a mos coun ably many eigen alues by Lemma 2.9, i ollows om Lemma 2.8 ha
he e exis a bi a ily la ge eal numbe s Nsuch ha bo h ZΦand ZΨa e e godic
unde he map TN. By a change o coo dina es we may suppose ha Nis an in ege .
Le ε >0and N0∈N. Combining Ego o ’s heo em wi h he well-known ac ha
dim η≤lim in N→∞ 1
NHN(η) o any p obabili y measu e η, Bi khoff’s e godic heo em
applied o TN o some N≥N0asse s ha o λΨ-a.e. (j, θ, , ),
lim
n→∞
1
n
n
k=1
1
N˜
HN(G(TkN (j,θ, , ))) >min{1,dim ν}−ε/2.
By con inui y o
˜
HN, his ac ually holds o e e y 0 ≤ <g(j, θ). Mo eo e , we can
eplace (j, θ, , )by (j, θ(j), 1, 0) since he o bi s o θand θ(j) unde TNa e asymp o ic
by Lemma 2.3. Thus, ecalling (6.14), we ha e o ¯ν-almos e e y j ∈ΛN ha
lim
n→∞
1
n
n
k=1
1
N˜
HN(πθ(j)νj|ik)>min{1,dim ν}−ε/2.(6.16)
Repea ing he exac same a gumen o μ, we also find ha o ¯μ-almos e e y i ∈ΓN,
52 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
lim
n→∞
1
n
n
k=1
1
N˜
HN(πθ(i)μi|ik)>min{1,dim μ}−ε/2 (6.17)
Now we wish o es ima e he en opy o μi|ik∗νj|ikusing (6.15)and he es ima es
(6.17)and (6.16). Howe e , o ha we need a lowe bound o he angle be ween θ(i)
and θ(j).
By Lemma 2.4 and Fubini, d(θ(i), θ(j)) >0 o ¯μׯν-almos e e y (i, j). Fo e -
e y n ∈N, le Xn={(i, j) ∈ΓN×ΛN:d(θ(i), θ(j)) ≥1/n}, and no e ha
¯μׯν(n∈NXn) =1. Fix now an in ege n, and no e ha o e e y (i, j) ∈Xn,
πθ(j)⊥|θ(i)=R−1
θ(j)⊥Rθ(i)cos d(θ(i), θ(j)⊥)πθ(i)whe e cos d(θ(i), θ(j)⊥) ≥1/2n. I ol-
lows om Lemmas 2.18 and 2.17 ha
˜
HN(πθ(i)μi|ik∗πθ(j)νj|ik)
≥˜
HN(S∗
log cos d(θ(i),θ(j)⊥)πθ(i)μi|ik)+ ˜
HN(πθ(i)μi|ik∗πθ(j)νj|ik|πθ(j)⊥)−O(1)
≥˜
HN+log cos d(θ(i),θ(j)⊥)(πθ(i)μi|ik)+ ˜
HN(πθ(j)νj|ik|πθ(j)⊥)−O(1)
=˜
HN+log cos d(θ(i),θ(j)⊥)(πθ(i)μi|ik)+ ˜
HN(πθ(j)νj|ik)−O(1)
o e e y (i, j) ∈Xnand k∈N, whe e he las equali y ollows om he ac ha
πθ(j)νj|ikis suppo ed on he line θ(j). Di iding by Nand using (6.17), (6.16), (6.15)
and he ac ha cos d(θ(i), θ(j)⊥) ≥1/2nwe ob ain ha o ¯μׯν-a.e. (i, j) ∈Xn,
lim
n→∞
1
n
n
k=1
1
N˜
HN(μi|ik∗νj|ik)≥min{1,dim μ}+min{1,dim ν}−ε,
as long as Nis la ge enough wi h espec o εand n. Since ¯μׯν(n∈NXn) =1, his
comple es he p oo .
P oo o Claim 4.3.Le ε >0, and le Φdeno e he conjuga ed IFS as in he s a emen o
he claim. Le μbe a ully suppo ed sel -affine measu e associa ed o Φwi h dim μ >1.
Once again, i will be con enien o eplace HNby he con inuous subs i u e
˜
HNdefined
in he p oo o Claim 4.2. Fo ¯μ×μFa.e. (i, θ)we ha e
lim
N→∞
1
N˜
HN(μi,θ)=dimμi,θ =dimμ−1
by Theo em 2.7.
Recall he defini ion o he unc ion F:ZΦ→P(R2) om (6.6), and ha F:Z
Φ→
P(R2)is defined by F(i, θ, ) =(i, θ, 1, ). Since μi,θ is a.s. exac dimensional and non-
a omic, by a sligh modifica ion o he p oo o Lemma 2.19 he e exis s N0∈Nand a
se S⊆Z
Φwi h λ
Φ(S) ≥1 −εsuch ha
1
N˜
HN((F(i,θ, ))I)≤dim μ−1+ε(6.18)
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 53
o e e y (i, θ, ) ∈S, e e y N≥N0and e e y in e al Ion he y-axis such ha 0 ∈I
and |I| =2. Since he unc ion → 1
N˜
HN((F(i, θ, ))I)is con inuous o ¯μ×μF-almos
e e y (i, θ), (6.18)in ac holds o e e y 0 ≤ ≤ (i, θ).
By P oposi ion 6.1, we may choose Nso ha Z
Φis e godic unde he disc e e- ime
map TN. Bi khoff’s e godic heo em asse s ha o λ
Φ-almos e e y (i, θ, ),
lim sup
n→∞
1
n
n
k=1
1
N˜
HN((F(T(k−1)N(i,θ, )))Ik)≤dim μ−1+ε
o any in e als (Ik)n
k=1 on he y-axis such ha 0 ∈Ikand |Ik| =2. As easoned abo e,
his in ac holds o e e y ≥0. Mo eo e , since he unc ions Fand Fdiffe only by
a eflec ion which does no affec en opy, o ¯μ×μF-almos e e y (i, θ)we ha e
1
n
n−1
k=m
1
N˜
HN(F(i,θ,1,(k−1)N−C(i,θ))Ik)
=1
n
n−1
k=m
1
N˜
HN((F(i,θ,(k−1)N−C(i,θ)))Ik)
≤dim μ−1+ε
o la ge enough n, whe e C(i, θ)is he cons an o Lemma 6.4 and m ∈Nis such ha
(m −1)N−C(i, θ) ≥0. Finally, using (6.8)and fixing he choice o (Ik)k, we ob ain
lim sup
n→∞
1
n
n−1
k=0
1
N˜
HN(π2μDkN (Π(i)))≤dim μ−1+2ε
o ¯μ-almos e e y i ∈ΓN.
P oo o Claim 4.4.Le ε >0, and deno e by Φand Ψ he conjuga ed IFSs as in he
s a emen . Le μand νbe ully suppo ed sel -affine measu es associa ed o Φand Ψ.
Le
˜
HNdeno e he con inuous subs i u e o HNas in he p oo o Claim 4.2. Recall he
defini ions o he unc ions F:ZΦ→P(R2)and G :ZΨ→P(R2) om (6.6)and (6.9).
By Ma s and’s p ojec ion heo em, i τand κa e any exac -dimensional Bo el measu es
on he line, hen o Lebesgue-almos e e y ≥0we ha e
dim(S τ∗κ)=min{1,dim τ+dimκ}.
Combining his wi h Theo ems 2.6 and 2.7, we find ha
dim(F(i,θ
1,u,
1)∗G(j,θ
2, ,
2)) = min{1,dim μ−1+dimν}(6.19)
54 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
o λΦ×λΨ-almos e e y (i, θ1, u, 1, j, θ2, , 2). Since uand a e d awn om he
uni o m measu e on {−1, 1}, (6.19)holds o λ
Φ×λ
Ψ-almos e e y (i, θ1, 1, j, θ2, 2)
and e e y (u, ) ∈{−1, 1}2
Now, o ¯μ×μF-a.e. (i, θ), any in e al ha con ains he o igin has posi i e μi,θ-
measu e. The e o e, o any ε >0, applying Lemma 2.19, Ego o ’s heo em and (6.19)
gi es a se S⊆Z
Φ×Z
Ψwi h λ
Φ×λ
Ψ(S) >1 −εand an in ege N0such ha
1
N˜
HN(F(i,θ
1,u,
1)I∗G(j,θ
2, ,
2)) ≥dim(F(i,θ
1,u,
1)∗G(j,θ
2, ,
2)) −ε
≥min{1,dim μ−1+dimν}−ε
o e e y N≥N0, (i, θ1, 1, j, θ2, 2) ∈S, (u, ) ∈{−1, 1}2and any in e al Isuch ha
0 ∈Iand |I| =2. Since he ope a ions (·)Iand (·)Idiffe only by a ansla ion when
|I| =2, i ollows om Lemma 2.16 ha also
1
N˜
HN(F(i,θ
1,u,
1)I∗G(j,θ
2, ,
2)) ≥min{1,dim μ−1+dimν}−ε
o e e y N≥N0, (i, θ1, 1, j, θ2, 2) ∈S, (u, ) ∈{−1, 1}2and any in e al Isuch ha
0 ∈Iand |I| =2.
P oposi ion 6.1 asse s ha he in ege Ncan be chosen o ha Z
Φ×Z
Ψis e godic
unde TN. I ollows om Bi khoff’s e godic heo em ha o λ
Φ×λ
Ψ-almos e e y
(i, θ1, 1, j, θ2, 2), limn→∞ 1
nn
k=1 1S(TkN (i, θ1, 1, j, θ2, 2)) ≥1 −εand consequen ly,
lim in
n→∞
1
n
n
k=1
1
N˜
HN(F(TkN (i,θ
1,1,
1))Ik∗G(TkN (j,θ
2,1,
2)))
≥min{1,dim μ−1+dimν}−2ε. (6.20)
A guing as in he p oo s o Claims 4.2 and 4.3 and using con inui y o con olu ion, we
see ha he abo e holds o e e y 0 ≤ 1≤ (i, θ1)and 0 ≤ 2≤g(i, θ2).
No e ha G(j, θ2, 1, 2)is con inuous in θ2, uni o mly o e (j, θ2, 2) ∈ZΨ, and ha
o ¯ν×ν∗
F-almos e e y (j, θ) ∈ΛN×RP1we ha e d(B∗
j|kθ2, B∗
j|k0⊥) →0as k→∞,
by Lemma 2.3. The e o e, we may eplace θ2by 0⊥so ha (6.20) s ill holds. Finally,
applying (6.8)and (6.14) o (6.20)wi h p ope choices o 1and 2, we ob ain
lim in
n→∞
1
n
n−1
k=0
1
N˜
HN(π2μDkN (Π(i)) ∗π2νj|ik)≥min{1,dim μ−1+dimν}−6ε
o ¯μ-almos e e y i ∈ΓNand ¯ν-almos e e y j ∈ΛN.
Re e ences
[1] B. Bá ány, On he Led appie -Young o mula o sel -affine measu es, Ma h. P oc. Camb. Philos.
Soc. 159 (3) (2015) 405–432.
A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770 55
[2] B. Bá ány, M. Hochman, A. Rapapo , Hausdo ff dimension o plana sel -affine se s and measu es,
In en . Ma h. 216 (3) (2019) 601–659.
[3] B. Bá ány, A. Käenmäki, Led appie -Young o mula and exac dimensionali y o sel -affine measu es,
Ad . Ma h. 318 (2017) 88–129.
[4] B. Bá ány, A. Käenmäki, I.D. Mo is, Domina ion, almos addi i i y, and he modynamic o malism
o plana ma ix cocycles, Is . J. Ma h. 239 (1) (2020) 173–214.
[5] B. Bá ány, A. Käenmäki, A. Pyö älä, M. Wu, Scaling limi s o sel -con o mal measu es, 2023.
P ep in , a ailable a a Xi :2308 .11399.
[6] B. Bá ány, A. Käenmäki, E. Rossi, Assouad dimension o plana sel -affine se s, T ans. Am. Ma h.
Soc. 374 (2) (2021) 1297–1326.
[7] B. Bá ány, A. Käenmäki, H. Yu, Fine geome y o plana sel -affine se s, 2021. P ep in , a ailable
a a Xi :2107 .00983.
[8] Y. Benois , J.-F. Quin , Random Walks on Reduc i e G oups, E gebnisse de Ma hema ik und Ih e
G enzgebie e. 3. Folge. a Se ies o Mode n Su eys in Ma hema ics [Resul s in Ma hema ics and
Rela ed A eas. 3 d Se ies. a Se ies o Mode n Su eys in Ma hema ics], ol. 62, Sp inge , Cham,
2016.
[9] J. Bochi, N. Gou melon, Some cha ac e iza ions o domina ion, Ma h. Z. 263 (1) (2009) 221–231.
[10] P. Bouge ol, J. Lac oix, P oduc s o Random Ma ices wi h Applica ions o Sch ödinge Ope a o s,
P og ess in P obabili y and S a is ics, ol. 8, Bi khäuse Bos on, Inc., Bos on, MA, 1985.
[11] R. Bowen, Equilib ium S a es and he E godic Theo y o Anoso Diffeomo phisms, Lec u e No es
in Ma hema ics, ol. 470, Sp inge -Ve lag, Be lin-New Yo k, 1975.
[12] C. B uce, X. Jin, Fu s enbe g sumse conjec u e and Mandelb o pe cola ions, 2022. P ep in , a ail-
able a a Xi :2211 .16410.
[13] T.M. Co e , J.A. Thomas, Elemen s o In o ma ion Theo y, second edi ion, Wiley-In e science [John
Wiley & Sons], Hoboken, NJ, 2006.
[14] E.B. Da ies, Spec al Theo y and Diffe en ial Ope a o s, Camb idge S udies in Ad anced Ma he-
ma ics, ol. 42, Camb idge Uni e si y P ess, Camb idge, 1995.
[15] N. Edeko, M. Ge lach, V. Kühne , Measu e-p ese ing semiflows and one-pa ame e Koopman semi-
g oups, Semig oup Fo um 98 (1) (2019) 48–63.
[16] K.J. Falcone , F ac al Geome y, John Wiley & Sons, L d., Chiches e , 1990, Ma hema ical oun-
da ions and applica ions.
[17] A.-H. Fan, K.-S. Lau, H. Rao, Rela ionships be ween diffe en dimensions o a measu e, Mona she e
Ma h. 135 (3) (2002) 191–201.
[18] D.-J. Feng, Dimension o in a ian measu es o affine i e a ed unc ion sys ems, Duke Ma h. J.
172 (4) (2023) 701–774.
[19] A. Fe guson, J.M. F ase , T. Sahls en, Scaling scene y o (×m, ×n) in a ian measu es, Ad . Ma h.
268 (2015) 564–602.
[20] J.M. F ase , The Fou ie spec um and sumse ype p oblems, Ma h. Ann. (2024).
[21] H. Fu s enbe g, Noncommu ing andom p oduc s, T ans. Am. Ma h. Soc. 108 (1963) 377–428.
[22] B.C. Hall, Quan um Theo y o Ma hema icians, G adua e Tex s in Ma hema ics, ol. 267, Sp inge ,
New Yo k, 2013.
[23] M. Hochman, Dynamics on ac als and ac al dis ibu ions, 2010. P ep in , a ailable a a Xi :
1008 .3731.
[24] M. Hochman, Geome ic igidi y o ×min a ian measu es, J. Eu . Ma h. Soc. 14 (5) (2012)
1539–1563.
[25] M. Hochman, No es on E godic Theo y, Heb ew Uni e si y o Je usalem, 2013, A ailable a h ps://
ma h .huji .ac .il /mhochman /cou ses /e godic - heo y -2012 /no es .final .pd .
[26] M. Hochman, On sel -simila se s wi h o e laps and in e se heo ems o en opy, Ann. Ma h. (2)
180 (2) (2014) 773–822.
[27] M. Hochman, On sel -simila se s wi h o e laps and in e se heo ems o en opy in Rd. 2015,
P ep in , a ailable a a Xi :1503 .09043.
[28] M. Hochman, A. Rapapo , Hausdo ff dimension o plana sel -affine se s and measu es wi h o e -
laps, J. Eu . Ma h. Soc. (2021), To appea , a ailable a a Xi :1904 .09812.
[29] M. Hochman, P. Shme kin, Local en opy a e ages and p ojec ions o ac al measu es, Ann. Ma h.
(2) 175 (3) (2012) 1001–1059.
[30] A. Käenmäki, H. Koi usalo, E. Rossi, Sel -affine se s wi h fib ed angen s, E god. Theo y Dyn.
Sys . 37 (6) (2017) 1915–1934.
[31] A. Käenmäki, T. Sahls en, P. Shme kin, S uc u e o dis ibu ions gene a ed by he scene y flow,
J. Lond. Ma h. Soc. (2) 91 (2) (2015) 464–494.
56 A. Pyö älä / Ad ances in Ma hema ics 451 (2024) 109770
[32] T. Kemp on, The scene y flow o sel -affine measu es, 2015. P ep in , a ailable a a Xi :1505 .01663.
[33] C.G.T.d.A. Mo ei a, Sums o egula Can o se s, dynamics and applica ions o numbe heo y, in:
Miskolc, 1998, in: In e na ional Con e ence on Dimension and Dynamics, ol. 37, 1998, pp. 55–63.
[34] F. Naza o , Y. Pe es, P. Shme kin, Con olu ions o Can o measu es wi hou esonance, Is . J.
Ma h. 187 (2012) 93–116.
[35] W. Pa y, M. Pollico , Ze a unc ions and he pe iodic o bi s uc u e o hype bolic dynamics,
As é isque 268 (1990) 187–188.
[36] Y. Pe es, P. Shme kin, Resonance be ween Can o se s, E god. Theo y Dyn. Sys . 29 (1) (2009)
201–221.
[37] E. Rossi, Visible pa o domina ed sel -affine se s in he plane, Ann. Fenn. Ma h. 46 (2) (2021)
1089–1103.
[38] E. Rossi, P. Shme kin, On measu es ha imp o e Lqdimension unde con olu ion, Re . Ma .
Ibe oam. 36 (7) (2020) 2217–2236.
[39] K. Schmid , Cocycles on E godic T ans o ma ion G oups, Macmillan Lec u es in Ma hema ics,
ol. 1, Macmillan Co. o India, L d., Delhi, 1977.
[40] P. Wal e s, An In oduc ion o E godic Theo y, G adua e Tex s in Ma hema ics, ol. 79, Sp inge -
Ve lag, New Yo k, 1982.