Blu o he Th ee–Body P oblem
Fini e Families unde Coa se Obse a ion
Aleksanda Pe išić
Decembe 2025
Abs ac
We ea he New onian h ee–body p oblem as a es bed o he gene al p inciple ha
unde blu he e a e only ini ely many e ec i e amilies o ajec o ies. We in oduce blu
ope a o s on bo h he phase space and he pa ame e space (masses, g a i a ional cons an ),
and de ine blu – amilies o o bi s: classes o solu ions whose blu ed ajec o ies emain
uni o mly close o e a ixed ime window. On any compac , non–collision ene gy egion
and ini e ime in e al, we show ha a ini e se o blu – amilies su ices o app oxima e
all ajec o ies a a gi en esolu ion. When pa ame e s a e blu ed as well, we ob ain ini e
amilies in he join pa ame e –s a e space, and we desc ibe blu –s able dynamical egimes
whe e quali a i e o bi ypes (bounded/escape) a e cons an on blu –classes.
Analy ically, he esul s a e s aigh o wa d applica ions o con inuous–dependence es i-
ma es. Concep ually, hey i in o a b oade “blu s. ini eness” pic u e ha also appea s in
numbe – heo e ic p oblems (e.g. Colla z ce i ica es): one ades sha p, global knowledge
o a ini e numbe o coa se ep esen a i es. The h ee–body case p o ides a conc e e, i-
ni e–dimensional illus a ion o how blu u ns chao ic dynamics in o ini ely many obse able
his o ies.
Con en s
1 In oduc ion 2
2 The New onian h ee–body p oblem 2
2.1 Phase space and ec o ield .............................. 2
2.2 Non–collision ene gy shells ............................... 3
3 Blu on phase space and blu – amilies 3
3.1 Phase–space blu .................................... 3
3.2 Blu – amilies on a compac ene gy shell ....................... 4
4 Blu ing pa ame e s as well 5
4.1 Pa ame e space and con inui y ............................ 5
4.2 Blu on pa ame e space ................................ 6
4.3 Join blu – amilies in pa ame e and s a e ...................... 6
5 Blu –s able dynamical egimes 7
5.1 Coa se o bi ypes ................................... 7
6 Blu wi h an Obse e : Fini e Obse able His o ies 9
6.1 Obse e , measu emen map, and obse a ion me ic ................ 9
6.2 Obse able blu –dis ance and ini e obse able his o ies .............. 9
6.3 Insol abili y and he ole o i educible blu ..................... 10
1
7 Discussion and ou look 11
1 In oduc ion
The New onian h ee–body p oblem is a classical labo a o y o chao ic dynamics and s uc u al
complexi y. E en in he simples case o h ee poin masses mo ing unde mu ual New onian
a ac ion, he long– ime beha iou o o bi s exhibi s ex eme sensi i i y o ini ial condi ions
and pa ame e s.
F om a blu pe spec i e, his sensi i i y is a ea u e, no a bug. Ra he han asking o exac
poin wise p edic ion, we ask:
Gi en a blu scale
in phase space (and possibly in pa ame e space) and a ole ance
ε
on a ini e ime ho izon [0
, T
], how many dis inc blu – amilies o ajec o ies does
he sys em eally ha e?
The answe is: ini ely many. Mo e p ecisely:
• ix a compac non–collision ene gy egion and a ime ho izon T;
•
ix blu scales
(s a e space) and
σ
(pa ame e s), and a ole ance
ε
o he blu ed
ajec o ies;
•
hen he e exis s a ini e se o ep esen a i e ajec o ies such ha e e y (pa ame e , ini ial
condi ion) pai in he egion is blu –close o one o hese ep esen a i es on [0, T ].
This is almos au ological om he iewpoin o con inuous dependence: he low is
con inuous in ini ial da a and pa ame e s; on a compac se we ha e uni o m con inui y;
he e o e, ini ely many sample poin s su ice a any ixed esolu ion.
The poin o his no e is o:
(i) o mula e his in an explici blu language, including blu on pa ame e s;
(ii)
s a e anspa en ini eness esul s (blu – amilies and blu –s able egimes) ha can be
ansplan ed o mo e complica ed sys ems;
(iii)
isola e he pa e n: ini e amilies unde blu as a kind o coa se ce i ica e, echoing he
ini e ce i ica es used in o he con ex s (e.g. Colla z Lyapuno p oo s).
The esul s he e a e delibe a ely modes on he analy ic side: no hing abou global in eg a-
bili y, no egula i y mi acle. The con ibu ion is a clean blu o maliza ion in a e y conc e e,
classical model.
2 The New onian h ee–body p oblem
2.1 Phase space and ec o ield
We wo k in h ee spa ial dimensions; he plana case is a simpli ica ion bu does no change he
blu logic.
Le
qi∈R3
be he posi ion and
pi∈R3
he momen um o he
i
– h body,
i
= 1
,
2
,
3. Deno e
q= (q1, q2, q3)∈R9, p = (p1, p2, p3)∈R9, z = (q, p)∈X:= R18.
Le
m
= (
m1, m2, m3
)
∈
(0
,∞
)
3
be he masses and
G >
0 he g a i a ional cons an . The
Hamil onian is
HG,m(q, p) =
3
X
i=1
|pi|2
2mi
−GX
1≤i<j≤3
mimj
|qi−qj|.(1)
2
The equa ions o mo ion a e Hamil on’s equa ions
˙z=FG,m(z)=J∇HG,m(z),
wi h
J
he s anda d symplec ic ma ix. Fo now we ix
G
and
m
and w i e
F
=
FG,m
,
H
=
HG,m
.
2.2 Non–collision ene gy shells
Le Cbe he collision se :
C=[
1≤i<j≤3
{z= (q, p):qi=qj}.
We es ic a en ion o a compac , non–collision egion in phase space.
De ini ion 2.1 (Non–collision compac egion).Fix ene gy le el
E∈R
, adius
R >
0and a
collision–a oidance ma gin δ > 0. De ine
ΣE,R,δ =z= (q, p)∈R18 :H(z)=E, max
i|qi|≤R, max
i|pi|≤R, |qi−qj| ≥ δ∀i < j.
P oposi ion 2.2 (Lipschi z ec o ield on Σ
E,R,δ
).On Σ
E,R,δ
he ec o ield
F
is
C∞
and
Lipschi z: he e exis s L=L(E, R, δ, G, m)>0such ha
∥F(z)−F(z′)∥≤L∥z−z′∥ ∀z, z′∈ΣE,R,δ.
Ske ch.
On Σ
E,R,δ
all mu ual dis ances
|qi−qj|
a e bounded below by
δ
, so he po en ial
−Gmimj/|qi−qj|
and i s de i a i es a e bounded. The kine ic pa is polynomial. Hence
∇H
is smoo h wi h bounded i s de i a i es on ΣE,R,δ, which implies he Lipschi z bound.
By s anda d ODE heo y, o each
z0∈
Σ
E,R,δ
he e exis s a unique solu ion
z
(
;
z0
)de ined
on some in e al a ound 0. Since Σ
E,R,δ
is compac and
F
is Lipschi z on i , he e is a uni o m
exis ence ime
T∗
=
T∗
(
E, R, δ, G, m
)such ha solu ions s a ing in Σ
E,R,δ
emain in Σ
E,R,δ
o
| | ≤ T∗. Fo T≤T∗we deno e he low by
Φ : ΣE,R,δ →ΣE,R,δ,Φ (z0)=z( ;z0).
P oposi ion 2.3 (Con inuous dependence on ini ial da a).Fo
| |≤T
wi h
T≤T∗
and
z0, z′
0∈ΣE,R,δ,
∥Φ (z0)−Φ (z′
0)∥≤eLT ∥z0−z′
0∥,
whe e Lis he Lipschi z cons an om P oposi ion 2.2.
P oo .
G onwall’s inequali y applied o
˙z
=
F
(
z
)and he Lipschi z es ima e yields he s anda d
bound ∥Φ (z0)−Φ (z′
0)∥≤eL| |∥z0−z′
0∥.
3 Blu on phase space and blu – amilies
We now in oduce blu on he phase space and de ine blu – amilies o ajec o ies.
3.1 Phase–space blu
Since X=
R18
is ini e–dimensional, many blu ope a o s a e a ailable. Fo de ini eness we use a
Gaussian con olu ion blu on unc ions, and he associa ed ac ion on ajec o ies.
3
De ini ion 3.1 (Phase–space blu ope a o ).Fo
>
0, le
κ
:X
→
[0
,∞
)be a Gaussian
ke nel
κ (z) = 1
(2π 2)9exp−∥z∥2
2 2.
Fo a bounded con inuous obse able φ:X→Rde ine i s blu ed e sion
(B φ)(z) = ZX
κ (z−z′)φ(z′)dz′.
Fo a ajec o y z( )in Xwe de ine he blu ed ajec o y a scale as he cu e
7→ (B δz( )),
o mo e conc e ely ia any ixed obse able
φ
: we ack
7→
(B
φ
)(
z
(
)) ins ead o
7→ φ
(
z
(
)).
Fo ou ini eness esul s i is enough o wo k a he le el o he s a e i sel and iew blu as
a bounded linea map B :X→Xsa is ying:
•∥B ∥ ≤ 1 o all ;
•B →Id s ongly as ↓0.
One may hink o B
as p ojec ing on o a coa se g id o mesh
o on o low Fou ie modes; he
p ecise choice does no a ec he logic.
De ini ion 3.2 (Blu dis ance on ajec o ies).Fix
>
0and a ime ho izon
T >
0. Fo wo
ajec o ies z( ),˜z( )de ined on [0, T]we de ine hei blu ed dis ance
d ,T (z, ˜z) := sup
0≤ ≤T
∥B z( )−B ˜z( )∥.
3.2 Blu – amilies on a compac ene gy shell
We now de ine blu – amilies on ΣE,R,δ.
De ini ion 3.3 ((
, ε, T
)–blu – amily).Le
K⊂
Σ
E,R,δ
be compac . Fix a blu scale
>
0,
ole ance
ε >
0and a ime ho izon
T≤T∗
. A amily o e e ence ini ial da a
{zj}N
j=1 ⊂K
is
called an ( , ε, T)–blu – amily o Ki o e e y z0∈K he e exis s some jsuch ha
d ,T Φ·(z0),Φ·(zj)= sup
0≤ ≤T
∥B Φ (z0)−B Φ (zj)∥≤ε.
The co esponding blu ed ajec o ies 7→ B Φ (zj)a e called blu – amily ep esen a i es.
In wo ds: wi hin he blu esolu ion
and ole ance
ε
, e e y ajec o y s a ing in
K
is
indis inguishable om one o he ep esen a i e ajec o ies up o ime T.
The main obse a ion is ha such amilies always exis and a e ini e.
Theo em 3.4 (Fini e blu – amilies on a compac egion).Le Σ
E,R,δ
and
T≤T∗
be as abo e,
and le
K⊂
Σ
E,R,δ
be compac . Fo any blu scale
>
0and ole ance
ε >
0 he e exis s a ini e
( , ε, T)–blu – amily o K.
Mo e p ecisely, one may choose e e ence poin s {zj}N
j=1 such ha K⊂SN
j=1 B(zj, δ0)wi h
δ0=ε e−LT ,
whe e
L
is he Lipschi z cons an o
F
on Σ
E,R,δ
, and
N
is he co e ing numbe o
K
by balls
o adius δ0.
4
P oo . Le Lbe he Lipschi z cons an om P oposi ion 2.2. By P oposi ion 2.3,
∥Φ (z0)−Φ (z′
0)∥≤eLT ∥z0−z′
0∥
o all 0≤ ≤Tand z0, z′
0∈K. Since ∥B ∥ ≤ 1,
∥B Φ (z0)−B Φ (z′
0)∥≤∥Φ (z0)−Φ (z′
0)∥≤eLT ∥z0−z′
0∥.
Choose a ini e
δ0
–ne
{zj}N
j=1
in
K
wi h
δ0
=
εe−LT
; such a ne exis s because
K
is compac .
Then o any z0∈K he e is a jwi h ∥z0−zj∥≤δ0, and he e o e
d ,T Φ·(z0),Φ·(zj)≤eLT δ0=ε.
So
{zj}
is an (
, ε, T
)–blu – amily. The bound on
N
is immedia e om he co e ing de ini ion.
The heo em con i ms he in ui i e pic u e: on a compac , non–collision ene gy egion and
ini e ime ho izon, he e a e only ini ely many blu – amilies a any ixed blu and ole ance. As
we e ine blu (smalle
o
ε
), he equi ed numbe o amilies g ows wi h he co e ing numbe
o Kand exponen ially wi h he ime window T( h ough eLT ).
Rema k 3.5 (Dependence on blu scale).No e ha he bound o
N
in Theo em 3.4 does
no depend explici ly on
: we did no use ha B
is app oxima ing he iden i y as
→
0. To
inco po a e ha , one may le he ole ance
ε
=
ε
(
)depend on
, sh inking as
→
0, o e lec
he ac ha coa se blu allows la ge ole ance in he sha p no m; he exis ence esul hen
adap s wi h a escaled δ0( ).
4 Blu ing pa ame e s as well
The h ee–body low depends no only on he ini ial condi ion bu also on pa ame e s (
G, m1, m2, m3
).
We now in oduce blu on pa ame e space and ob ain ini eness in he join pa ame e –s a e
space.
4.1 Pa ame e space and con inui y
Le he pa ame e space be
P= [Gmin, Gmax]×[m1,min, m1,max]×[m2,min, m2,max]×[m3,min, m3,max],
wi h all bounds posi i e and ini e. Fo
p
= (
G, m1, m2, m3
)
∈
Pwe w i e
Fp
and Φ
p
o he
ec o ield and low associa ed o ha pa ame e uple.
P oposi ion 4.1 (Uni o m Lipschi z and con inuous dependence on
p
).Le
K⊂
Σ
E,R,δ
be
compac and le Pbe as abo e, wi h all
mi
bounded away om 0and in ini y and
G
in a compac
in e al. Then:
(a) The e exis s LK>0such ha o all p∈P,
∥Fp(z)−Fp(z′)∥≤LK∥z−z′∥ ∀z, z′∈K.
(b)
Fo each
T≤T∗
, he low (
p, z0
)
7→
Φ
p
(
z0
)is uni o mly con inuous on P
×K
o 0
≤ ≤T
.
Ske ch.
On
K×
P, he masses and
G
a e bounded and bounded away om 0, and he posi ions
a e collision–sepa a ed by
δ
. The Hamil onian and i s de i a i es up o second o de a e bounded
uni o mly in (
p, z
), so he Lipschi z cons an in
z
can be chosen uni o mly in
p
. Fo (b),
Fp
depends smoo hly on
p
, so he solu ion depends con inuously on bo h
p
and
z0
; on a compac
domain P×Kand ini e ime in e al, his implies uni o m con inui y.
5
4.2 Blu on pa ame e space
We now de ine blu on pa ame e space analogously.
De ini ion 4.2 (Pa ame e blu ).Le B
(P)
σ
:P
→
Pbe a amily o blu ope a o s a scales
σ >
0, o ins ance gi en by con olu ion wi h a smoo h compac ly suppo ed ke nel on he box
P(ex ended by e lec ion a he bounda y), o by p ojec ion o a coa se g id o mesh
σ
. We
equi e:
•∥B(P)
σ∥≤1;
•B(P)
σ→Id as σ↓0.
We conside he join blu ope a o on pa ame e –s a e pai s (p, z)∈P×X:
Bσ, (p, z) := B(P)
σp, B z.
4.3 Join blu – amilies in pa ame e and s a e
We now de ine blu – amilies in he join space P×K.
De ini ion 4.3 ((
σ, , ε, T
)–join blu – amily).Fix compac
K⊂
Σ
E,R,δ
, pa ame e box P, blu
scales σ > 0, > 0, ole ance ε > 0and T≤T∗. A ini e collec ion o ep esen a i es
{(pj, zj)}N
j=1 ⊂P×K
is a (σ, , ε, T )–join blu – amily i o e e y (p, z0)∈P×K he e exis s jsuch ha
sup
0≤ ≤T
Bσ, p, Φp
(z0)−Bσ, pj,Φpj
(zj)
≤ε.
In wo ds: unde he obse e ’s blu on pa ame e s and phase space, e e y ealised h ee–body
con igu a ion o e [0
, T
]is indis inguishable om one o he ep esen a i e pa ame e – ajec o y
pai s.
Theo em 4.4 (Fini e join blu – amilies).Unde he assump ions o P oposi ion 4.1, o any
blu scales
σ >
0,
>
0, ole ance
ε >
0and ime ho izon
T≤T∗
, he e exis s a ini e
(σ, , ε, T)–join blu – amily o P×K.
Mo e conc e ely, he e exis s a ini e
δ
–ne
{
(
pj, zj
)
}N
j=1
in P
×K
such ha he abo e
p ope y holds, whe e
δ
=
δ
(
ε
)is de e mined by he uni o m con inui y modulus o (
p, z0
)
7→
(Bσ, p, Bσ, Φp
(z0)) on P×Kand [0, T].
P oo . By P oposi ion 4.1, he map
(p, z0, )7→ B(P)
σp, B Φp
(z0)
is con inuous on he compac se P
×K×
[0
, T
], hence uni o mly con inuous. Thus he e exis s
a modulus ω(·)such ha
∥(p, z0)−(p′, z′
0)∥ ≤ δ=⇒sup
0≤ ≤T
Bσ, p, Φp
(z0)−Bσ, p′,Φp′
(z′
0)
≤ω(δ).
Pick
δ >
0wi h
ω
(
δ
)
≤ε
, and choose a ini e
δ
–ne
{
(
pj, zj
)
}N
j=1
in P
×K
(possible by
compac ness). Then o any (
p, z0
) he e exis s
j
wi h
∥
(
p, z0
)
−
(
pj, zj
)
∥≤δ
, which gi es he
desi ed inequali y.
The heo em exp esses p ecisely he ini e– amilies–unde –blu in ui ion in he join pa ame e –
s a e space.
6
5 Blu –s able dynamical egimes
The ini eness heo ems abo e conce n ajec o ies as cu es. O en one is mo e in e es ed in
quali a i e o bi ypes: bounded mo ion, escape, collision, and e inemen s he eo (e.g. bina y
o ma ion, ejec ion o one body).
We ske ch how blu – amilies in e ac wi h coa se classi ica ion maps.
5.1 Coa se o bi ypes
Fix a classi ica ion ime ho izon Tmax >0. De ine a coa se o bi ype map
C:P×K→ {1,2,3},
whe e, say,
•
C(
p, z0
) = 1 i he o bi emains in a bounded egion (no escape, no collision) up o ime
Tmax;
•C(p, z0)=2i a collision occu s be o e Tmax;
•
C(
p, z0
)=3i some body escapes beyond a la ge ixed dis ance
Resc
be o e
Tmax
wi hou
collision.
The p ecise de ini ions a e no c ucial; he key poin is ha Cis a map om (
p, z0
) o a ini e
se o labels.
In gene al Cmay be highly discon inuous on P
×K
due o chao ic bounda ies be ween
egimes. Ne e heless, he e a e egions whe e i is locally cons an .
De ini ion 5.1 (Blu –s able egime).Le (
p∗, z∗
)
∈
P
×K
. We say ha (
p∗, z∗
)lies in a
blu –s able egime a scales (σ, )i he e exis s ε > 0such ha
C(p, z0)=C(p∗, z∗)
o all (p, z0)wi h
Bσ, (p, z0)−Bσ, (p∗, z∗)
≤ε.
In o he wo ds, wi hin he blu –class a ound (
p∗, z∗
) he coa se o bi ype is cons an ; he
egime is s able unde he obse e ’s blu .
P oposi ion 5.2 (Local blu –s abili y unde s uc u al s abili y).Suppose Cis locally cons an
in a (sha p) neighbou hood
U×V
o (
p∗, z∗
)in P
×K
. Then he e exis
σ0, 0>
0and
ε >
0
such ha (
p∗, z∗
)is blu –s able a all blu scales (
σ,
)wi h 0
< σ ≤σ0
,0
< ≤ 0
, wi h he
same label C(p∗, z∗).
P oo .
I Cis locally cons an nea (
p∗, z∗
), he e exis s
η >
0such ha C(
p, z0
) = C(
p∗, z∗
)
whene e
∥
(
p, z0
)
−
(
p∗, z∗
)
∥ ≤ η
. Choose
σ0, 0>
0so small ha
∥
B
σ,
(
p, z0
)
−
(
p∗, z∗
)
∥ ≤ η
whene e
∥
(
p, z0
)
−
(
p∗, z∗
)
∥≤η/
2and 0
< σ ≤σ0
,0
< ≤ 0
; his is possible since B
σ, →Id
as
σ, →
0. Then ake
ε
=
η/
2; he blu –ball o adius
ε
a ound B
σ,
(
p∗, z∗
)in blu ed no m
si s inside he sha p ball o adius ηa ound (p∗, z∗), whe e Cis cons an .
Combined wi h Theo em 4.4, his implies ha on any compac egion whe e Cis piecewise
locally cons an , he numbe o dis inc blu –s able o bi ypes a a gi en blu esolu ion is ini e.
Each blu – amily ep esen a i e in he join space can be labelled by i s o bi ype, and he
blu –s able egimes co espond o clus e s o ep esen a i es wi h he same label.
While his is s ill comple ely in he ealm o classical con inui y and s uc u al s abili y, he
blu language u ns i in o a clean s a emen : a any ini e esolu ion, he h ee–body sys em has
ini ely many blu –s able a es.
7
Concep ual akeaway: why blu is no op ional
F om a dis ance, blu may look like a epackaging o hings we al eady know: e o ba s,
coa se g ids, nume ical ole ances. The poin o he p esen analysis is ha , o genuinely
his o y–dependen sys ems such as he h ee–body p oblem, blu is no jus con enien , i is
ma hema ically una oidable.
A de e minis ic h ee–body sys em wi h exac pa ame e s and exac ini ial da a has a
unique his o y. In p ac ice, howe e , he pa ame e s and ini ial s a es li e in a small bu
i educible unce ain y egion: measu emen noise, modelling e o , un esol ed o ces, and
nume ical unca ion. Because he dynamics is s ongly sensi i e o small pe u ba ions, his
unce ain y egion does no emain small when anspo ed along he low; i s e ches, olds,
and h eads i sel h ough phase space in a way ha makes sha p his o ies meaningless beyond
a sho ho izon.
The blu iewpoin says: do no igh his, build i in o he de ini ion o he objec you a e
s udying. Fix om he s a a p ecise blu scale in pa ame e and s a e space— ha is, a egion
o unce ain y ha you accep as cons i u i e. Ins ead o one exac ajec o y, conside he
en i e cloud o ajec o ies compa ible wi h ha blu . Ou analysis shows ha , a any ixed blu
scale and any ixed ini e ime ho izon, his cloud b eaks in o only ini ely many blu ed amilies
o his o ies: equi alence classes o solu ions ha a e indis inguishable wi hin he accep ed e o
o all obse ables we a e willing o pay o . Inside each amily, he sys em s ill has genuine
mic oscopic eedom, bu ha eedom no longe p opaga es in o new mac oscopic scena ios; i
only changes which ep esen a i e o he same blu ed his o y we see.
In his sense, blu ades mic oscopic uniqueness o mac oscopic ini eness. The in ini e
his o y o he sys em is no a single agile cu e in phase space bu a compa ible sequence o
blu ed his o ies, one o each ime window and each esolu ion. Once he blu scale is hones ly
decla ed, he usual “pic u es” o he e olu ion (con igu a ions, ene gy exchanges, quali a i e
egimes) become s able objec s: hey a e ead as in a ian s o hese blu ed amilies a he han
as shadows o an una ainable exac ajec o y. Fa om being a weakness, his is p ecisely he
le el a which he h ee–body p oblem is obse able in he eal wo ld.
P inciple 5.3 (Conse a ion o blu budge ).Fix a sys em wi h low Φ
on a s a e space
X
,
and a blu scale
ha encodes an accep ed ini ial unce ain y (a small, hones chunk o missing
in o ma ion in pa ame e –s a e space). Suppose his unce ain y is modelled by a blu ope a o
B ha is anspo ed co a ian ly along he dynamics,
Φ
∗◦B ≈B ◦Φ
∗,
so ha we always look a he sys em h ough he same in o ma ional lens. Then he ini ial e o
does no blow up in he sense o eadabili y: he missing in o ma ion s ays o o de
o all
imes. Wha g ows is he in e nal complexi y inside each blu ed cell, no he blu budge i sel .
In mo e conc e e e ms: an ini ial e o ha is ea ed as a genuine educ ion o in o ma ion
inside a p ope in o ma ion ield does no , by i sel , make he sys em un eadable. Wha ails is
he Pla onic ideal o ze o e o . I we insis on acking a single sha p ajec o y, hen e en a iny
unknown agmen explodes unde a chao ic low and he desc ip ion becomes useless. Bu i we
ide oge he wi h he e o — ix om he s a a legi ima e blu scale and p opaga e i wi h he
dynamics— hen he in o ma ional gap emains bounded: 0
.
1bi s o missing in o ma ion emain
0.1bi s o missing in o ma ion, only ea anged.
The usual mis ake is o ake he i s ew ime s eps, whe e he sys em is almos sha p, as
he e e ence s anda d, and hen complain when ha le el o p ecision is los as he unknown
agmen s a s o a ec e e y hing else. F om he blu poin o iew, he co ec mo e is he
opposi e: accep he unce ain y a
= 0 as s uc u al, design he dynamics and obse ables so
ha blu is pushed exac ly whe e blu belongs, and hen ead he sys em consis en ly a ha
8
scale. The p ice o educing blu is ha mo e and mo e dis inc his o ies appea (e en ually
oo many o any compu a ional me hod), bu o any ixed, ame blu scale we e ain a ini e
collec ion o mac oscopic ou comes, each no mo e han he accep ed e o away om he “ ue”
his o y. Classical, poin like de e minism is eco e ed only as he singula limi whe e he blu
scale is d i en o ze o, and i is p ecisely in ha limi ha he desc ip ion becomes uns able.
6 Blu wi h an Obse e : Fini e Obse able His o ies
So a we ha e blu ed he h ee–body p oblem a he le el o phase space and pa ame e s,
wi hou explici ly modelling who is looking. In a Blu iche sky spi i , we now inse an obse e
and ein e p e he ini e blu – amilies as ini e se s o obse able his o ies.
6.1 Obse e , measu emen map, and obse a ion me ic
Le
O
deno e an obse e equipped wi h an ins umen model. We encode his as a measu emen
map
MO:P×X−→ Y,
whe e Yis a ( ini e–dimensional) da a space; ypical examples a e:
•Ya space o appa en posi ions and adial eloci ies (as ome ic + spec oscopic da a);
•Ya space o ligh cu es o ime–dependen luxes (pho ome ic da a);
•Ya ec o o de i ed quan i ies (pai wise dis ances, angles).
The map (
p, z
)
7→ MO
(
p, z
)includes he obse e ’s loca ion, line o sigh , p ojec ion e ec s, and
ins umen esponse.
We equip
Y
wi h a no m
∥·∥Y
(e.g. Euclidean on a ini e ec o o obse ables) and de ine
he obse a ion me ic on P×Xby
dO(p, z),(p′, z′):=
MO(p, z)−MO(p′, z′)
Y.
Thus wo pa ame e –s a e pai s a e close o
O
i and only i hey p oduce nea ly he same aw
da a a a single ins an .
We also allow blu a he le el o he da a: o a blu scale ρ > 0on Ywe le
B(Y)
ρ:Y→Y
be a linea ope a o wi h
∥
B
(Y)
ρ∥ ≤
1and B
(Y)
ρ→IdY
as
ρ↓
0(e.g. a con olu ion wi h a na ow
ke nel, o a p ojec ion o a coa se g id in da a space). Composing wi h ou ea lie blu on
pa ame e s and s a e we ob ain a o al obse ed blu
BO
σ, ,ρ(p, z) := B(Y)
ρMOB(P)
σp, B z.
6.2 Obse able blu –dis ance and ini e obse able his o ies
Fo a ixed obse e
O
, blu scales (
σ, , ρ
), and ime ho izon
T >
0, we de ine he obse able
blu –dis ance be ween wo pa ame e – ajec o y pai s (p, Φp
·(z0)) and (p′,Φp′
·(z′
0)) as
dO
σ, ,ρ;T(p, z0),(p′, z′
0):= sup
0≤ ≤T
BO
σ, ,ρp, Φp
(z0)− BO
σ, ,ρp′,Φp′
(z′
0)
Y.
9
