Simple and complex dynamics for circle maps

Publicacions Ma emá iques, Vol 37 (1993), 305-316 . Abs ac SIMPLE AND COMPLEX DYNAMICS FOR CIRCLE MAPS* LLUÍS ALSEDÁ AND VLADIMIR FEDORENKO The con inuous sel maps o a closed in e al o he eal line wi h ze o opological en opy can be cha ac e ized in e ms o he dy- namics o he map on i s chain ecu en se . In his pape we ex end his cha ac e iza ion o con inuous sel maps o he ci cle . We show ha , o hese maps, he chain ecu en se can exhibi a new dynamic beha iou which is speci ic o he ci cle maps o deg ee one . 1 . In oduc ion . The aim o his pape is o ex end he cha ac e iza ion o he complex and simple in e al maps (in he sense o posi i e o ze o opological en opy espec i ely) o ci cle maps . We shall s a by s a ing his cha - ac e iza ion o in e al maps o comple eness (see Theo em A) . To do i we ha e o in oduce he app op ia e no a ion . Le be a map om a opological space X in o i sel . We shall deno e by ' he map o o . . . o n imes (i n = 0 we se n = Id) . Le now be an in e al map ( ha is, a con inuous map om a closed in e al I o he eal line in o i sel ) . We say ha has a ho seshoe i he e exis n> 0 and wo closed in e als Il, 12 C I wi h pai wise disjoin in e io s such ha h U 12 C n (I1) and I l U 12 C n (I2 ) . The abo e condi ion was used o he i s ime by Sha ko skii(see [13]) and has been used widely in he s udy o in e al maps (see [9], [4] and [12]) . The name o ho seshoe was gi en o his condi ion by Misiu ewicz *This pape was w i en du ing a s age o he second au ho a he Cen e de Rece ca Ma emá ica . He was also pa ially suppo ed by he Fund o Fundamen al Resea ch o he S a e Commi ee o Uk aine on Science and Technology g an numbe 1/356 . The i s au ho has been pa ially suppo ed by he DGICYT g an numbe PB90- 0695 . 306  LL . ALSEDÁ, V . FEDORENKO in [9] . An in e al map ha ing a ho seshoe was called u bulen by Block (see [4]) and in [12] a simila no ion was called an L-scheme . We no e ha he se o in e al maps ha ing a ho seshoe is open and dense in he space o all in e al maps (we suppose his space endowed wi h he opology o he uni o m con e gen e) . Thus, in his sense, he p ope y ha a map has a ho seshoe is gene ic . I is well known ha a map has posi i e opological en opy i and only i i has a ho seshoe (see [9]) . In o he wo ds, he exis en e o ho seshoes cha ac e izes he complex in e al maps . Now we in oduce he necessa y no ions o cha ac e ize he simple in e al maps . Le S be a closed in a ian se o an in e al map . We say ha S spli s in o So and Sl i So and S I a e closed nonemp y subse s o S such ha (S,) n (S2) = 0 (whe e (Si) deno es he con ex hull o Si, i = 1, 2), Sl U S2 = S, (Sl) = S2 and (S2) =S i . We also say ha S spli s k imes i S spli s in o So and Sl and each o hem spli s (k -1) imes unde 2 . The se S is said o be simple i ei he i is a ixed poin o i spli s k imes o each k G 109 2 Ca d S (see [6]) . Rema k 1 .1 . F om he abo e de ini ion i ollows easily ha each simple se ei he consis s en a unique pe iodic o bi o does no con ain any pe iodic o bi . Recall ha i is a con inuous map om a me ic space X in o i sel he se o chain ecu en poin s o is deno ed by CR( ) and is de ined o be he se o all x E X such ha o each e > 0 he e exis s {xi}? o wi h xo = x,,, = x and ¡ (xi) - xi+11 G E o i = 0, 1, 2, . . . , n- 1 . The ollowing heo em cha ac e izes he complex and simple in e al maps (see [6]) . Theo em A . Each in e al map sa is ies one and only one o he ollowing wo condi ions . (a) has a ho seshoe . (b)  The chain ecu en se o is he union o all simple se s o . To ex end Theo em A o ci cle mapswe ha e o e o mula e he abo e no ions in his con ex . We shall ep esen he ci cle S 1 as he se {z E (C : Iz1 = 1} . Any con inuous map om S 1 in o i sel will be called a ci cle map . We no e ha he no ion o a simple se and o ho seshoe ex ends na u ally o ci cle maps by simply eplacing closed in e als by closed a es o he ci cle ( ha is, subse s o S l which a e homeomo phic o closed in e als o he eal line) . SIMPLE AND COMPLEX DYNAMICS FOR CIRCLE MAPS  30 7 We no e ha i an in e al map has a ho seshoe h, 12 hen we always ha e ha , o each i, j E {1, 2}, he e exis a closed in e al I~ C I i such ha n (I .~) = I, . Howe e , his is no he case i we a e alking abou ho seshoes o ci cle maps . Indeed, i h, 12 C S 1 is a ho seshoe o a ci cle map g i may happen ha gn (II) = S 1 in such a way ha gn ¡In (I 1 ) is injec i e and gn(a) = gn(b) E In (I 2 ) whe e aand b deno e he wo endpoin s o h . Then, clea ly, does no hold ha o each i, j E {1, 2} he e exis a closed a c Ij C I2 such ha gn (I .~) = h . Le be a map om a opological space X in o i sel and le x E X . We say ha x is a pe iodic poin o i n(x) = x o some n > 0 . The smalles n wi h he abo e p ope y is called he pe iod o x . I x E X is a pe iodic poin o o pe iod n hen he se {x, (X) .... , n -I (x)} will be called a pe iodic o bi o o pe iod n . In he sequel we shall deno e by Pe ( ) he se o pe iods o all pe iodic poin s o . Also, i x E X we shall deno e by w x ( ) he omega limi se o x which is de ined o be he se o all accumula ion poin s o { ' (x) : n >_ 0} . We will also use he no a ion w( ) o deno e UxEXwx ( ) . The mai i esul o his pape is he ollowing . Theo em 1 .2 . Each ci cle map sa is ies one and only one om he ollowing heee condi ions : (a) has a ho seshoe . (b)  The e exis n > 0 such ha w x ( n) is a simple se o each x E S' . (c) Pe ( ) = 0 . We no e ha , as o in e al maps, Condi ion (a) o he abo e heo em is gene ic in he space o ci cle maps endowed wi h he opology o he uni o m con e gen e and is a c i e ion o posi i e opological en opy (see [9]) . On he o he hand, i is known ha Condi ion (b) wi h n = 1, in he case o an in e al map, is equi alen o Condi ion (b) o Theo- em A (see o ins an e Theo em 2 o [7]) . Howe e , o ci cle maps i is no . Theo em 1 .2 is s a ed in his way o simplici y bu in Sec ion 4 he opological pic u e o he chain ecu en se in his case will be desc ibed in de ail . Finally, he dynamics o a ci cle map sa is ying Condi ion (c) o he abo e heo em can be oughly desc ibed as ollows . The map has a unique w-limi se which is minimal (Le . i has no closed in a ian p ope subse ) and he es ic ion o o i is semi-conjuga e o a o- a ion o he ci cle by an i a ional angle . A de ailed desc ip ion o he dynamics o such a map can be ound in [3] and [11] . 308  LL . ALSEDÁ, V . FEDORENKO 2 . De ini ions and p elimina y esul s . In his sec ion we will in oduce he necessa y no a ion o p o e The- o em 1 .2 . Also we will p o e a lemma ha will play a key ole in ha p oo . Le be a ci cle map . As usual, ins ead o wo king wi h i sel we shall use a li ing o . A con inuous map F : I[8 --> R is called a li ing o i e o F = o e, whe e e(x) = exp(27 ix) is he na u al p ojec ion om 1[8 o S 1 . We no e ha i F is a li ing o hen F + m is also a li ing o o each m E 7G and ha Fn is a li ing o ' . Also, he e exis s an in ege d such ha F(x + 1) = F(x) + d o each x E R . This numbe d is called he deg ee o and is deno ed by deg( ) . I is no di icul o see ha deg( n) = deg( )n . We say ha a poin x E R is pe iodic (mod . 1) o pe iod q o F i Fq(x) - x E 7L bu F 1 (x) - x ¢ 7L o j = 1, 2, . . . , q - 1 . Clea ly, x is a pe iodic (mod . 1) poin o F o pe iod q i and only i e(x) is a pe iodic poin o o pe iod q . Le F be a li ing o a ci cle map . In he sequel we shall deno e by Pe (F) he se o pe oods o all pe iodic (mod . 1) poin s o F . Clea ly, Pe ( ) = Pe (F) . Le be a ci cle map o deg ee one and le F be a li ing o . Fo x E R we de ine i s F- o a ion numbe as lim sup Fn (x) - x n - oo n and deno e i by PF(x) . We no e ha , since has deg ee 1, PF(x) = PF (X +M) o all m E 7G . Also, i x is a pe iodic (mod . 1) poin o pe iod q o F hen PF (x) = F9(x) - x E q The se {PF(x) : x E l[8} = {PF(x) : xE [0,1)} is deno ed by LF . I o in [8] p o ed ha LF is a closed in e al (pe haps degene a e o a poin ) o R . Thus, in he sequel LF will be called he o a ion in e al o F . The o a ion in e al o a li ing o a ci cle map o deg ee one cap u es a lo o i s dynamical p ope ies and plays a undamen al ole in hei s udy (see o ins ance [10] and [2]) . Le F be a li ing o a ci cle map o deg ee one . We de ine (see [1]) F . (x) = sup{F(y) : y < x} . SIMPLE AND COMPLEX DYNAMICS FOR CIRCLE MAPS  309 I is no di icul o see ha F u is non-dec easing and ha i is a li ing o a ci cle map o deg ee one . Mo eo e , F < Fu . Now we a e eady o s a e and p o e he lemma we a e looking o . Lemma 2 .1 . Le be a ci cle map and le F be a li ing o . Then one o he ollowing p ope ies hold : (a) has a ho seshoe . (b)  The e exis q E N,  p E 7G and . I = [x, x + 1]  C l[8 such ha (Fq _ p) (J) C J . (c) Pe ( ) = 0 . P oo . I deg( ) ¢ {-1, 0,1} hen Cea ly has a ho seshoe .  Thus, (a) holds . Assumenow ha deg( ) = 0 . Then, F(x + 1) = F(x) o all x E R . Hence, F(R) = F([0,1]) = [a, b] . I b <_ a + 1 hen we se q = 1, p= 0 and I = [a, a+ 1] ; and (b) holds . Thus, assume ha b > a+ l . Le c E R be such ha F (c) = a . Clea ly F(c+ 1) = a and he e exis d E (c, c+ 1) such ha F(d) = b . Se I l = e([c, d]) and 12 = e([d, c + 1]) . Clea ly Il and 12 a e a cs o S 1 and (I1) = (I2) DS 1 D I l U 12 . Thus, has a ho seshoe and (a) holds . Now we conside he case deg( ) = 1 . F om [10] i ollows ha Pe ( ) = 0 i and only i LF = {a} wi h a 0 Q and ha has a ho seshoe i LF is non-degene a e . Thus we only ha e o conside he case LF = {p/q} wi h q E N and p E 7G ela i ely p ime . F om [5] (see also [1, Theo em 3 .7 .20 and i s p oo ]) i ollows ha he e exis s x, a pe iodic (mod . 1) poin o F o pe iod q and o a ion numbe p/q, such ha Fi (x) = F,, (x) o all i > 0 . Se P={F'(x)+m :  i=0,1,2, . . .,q-1, mEZ}= = e  1({ 2(e(x)) :  i = 0,1, 2, . . . , q - 1}) = = { . . . x_2, x_1, xo, xl, X2 . . . . } wi h (xi, xi + 1)nP = 0 o all i E 7G . Se also G= Fq-p and G = Fú-p . Since F,, is non-dec easing so is G, 4 and, hence, G <_ G because F < F  . The e o e, i y E R, z E P and y < z hen G (y) < G . (y) < G . (z) = G (z) = z . Mo eo e , i is no di icul o see ha LG = {0} . Se J = [xo, xo + 1] .  We no e ha J C G(J) because G(xi) = xi o all i E 7L . Thus, G ¡ (J) C G'+1(J) o all i E N . Le K be he 31 0  LL . ALSEDÁ, V . FEDORENKO closu e o U °° 1 G'(J) . Clea ly K is a closed in e al such ha J C K C (-oo, xo+1] and G(K) C K . I K is no bounded, hen he e exis zE J and m E N such ha G'(z) < xo -1 . The e o e, he e exis s z E J such ha G'''(z) = z - 1 . Tha is, z is a pe iodic (mod . 1) poin o G wi h o a ion numbe -1/m . This con adic s he ac ha LG = {O} . Thus, K is o he o m [a, xo + 1] wi h a< xo . I a E P hen we ake I = [a, a+ 1] . Since a + 1 E P we ha e ha G(I) C (-oo, a+ 1] . On he o he hand, G(I) C G(K) C K . Thus G(I) C I and we a e done . Now assume ha a  P . Then he e exis s i < 0 such ha a E (xi, xi+l) . Since G(xi) = xi he e exis s a ixed poin o G in [xi, a] . Le y be he sup emum o he e ixed poin s . Since G(a) > a we see ha o each z E [y, ca] we ha e ha G(z) > y . Thus he se S = {y E [xi, ca] : G (y) = y and G(z) > y o each z E [y, a] } is non-emp y . So we se ,3 = in S and K = [,0, xo + l] . Since xi+1 E P and_xi+l <_ xo we ha e ha G([/3, ca]) C P, xi+1] C K . The e o e, G(K) C K . Now, i ,0 = xi we se I = [/ 0, 0+ 1] and we p oceed as abo e o ge G(I) C I . So we may assume ha 0 :7~ xi . I G([xi, i]) C [-00,3] hen we shall show ha G(I) C I wi h I = [,3, _ _( 3 ~- 1] . To see his i is enough o p o e ha i z < ~3 + 1 hen G(z) < ~3+1 because I C K and G(K) C K . We shall p o e i s ha i z <_ 0 hen G(z) < 0 . I z E [xi, N] hen his ollows by he assump ion . I z <_ xi hen, since xi E P we ha e ha G(z) <_ xi < ~ . Hence, since G is a li ing o e which has deg ee one, o each z <_  + 1 we ha e G(z) = G(z - 1) + 1 <_ ~3 + 1 and we a e done . Thus, we may assume ha he e exis s E [xi  C3] such ha G( ) >Ñ . Since xi =,L ~3, by he de ini ion o l, he e exis s zE [xi, ~3] such ha G(z) < xi . I <z hen he in e als [ , z] and [z,,(3] o m a ho seshoe o G . So F has a ho seshoe and (a) holds . I z < hen he e is a ixed poin o G in (z, ) . Le y be he sup emum o he e ixed poin s . We ha e y< <0 and G(z) > y  o all z I also G(z) >_ y o all z E [  3] hen y E S which con adic s he ac ha ~3 = in S . So, he e exis s 7E ( , ,0) such ha (5) = y . Then he in e als [y, ] and [ , ~] o m a ho seshoe o G . This ends he p oo o he lemma in he case when has deg ee one . Finally assume ha deg( ) = -l . By he de ini ion o deg ee o a map, has a ixed poin . Thus, Pe ( ) :,A 0 . Le us conside ' . I has SIMPLE AND COMPLEX DYNAMICS FOR CIRCLE MAPS  31 1 deg ee 1 and FZ as a li ing . Thus, in iew o he deg ee one case, ei he 2 has a ho seshoe o he e exis q E N, pE 7L and I = [x, x+1] such ha ((F2)q-p)(I) C I . Hence, ei he has a ho seshoe o (F 2 q-p)(I) C I . g Rema k 2 .2 . F om he p oo o he abo e lemma i ollows ha i is a ci cle map sa is ying (b) o Lemma 2 .1, hen deg( ) E {-1,0,1} . 3 . P oo o Theo em 1 .2 . We shall s a wi h some echnical lemmas on in e al maps . Lemma 3 .1 . Le be a con inuous map o he in e al I in o i sel ha ing a ho seshoe . Then, has a ho seshoe in In (I) . P oo .. Since has a ho seshoe he e exis n E N and h,12 C I, wo closed in e als wi h pai wise disjoin in e io s, such ha Il U12 C n(I1)  and  h U 12 C n (I2) . Then, he e exis J,', J2 C Il and Ji , J2 C 12, closed in e als wi h pai wise disjoin in e io s such ha Ji U J2 U Ji U JZ C 2n(J~z) wi h i, j E {1, 2} . Clea ly, wo o hese in e als a e con ained in In (I) . Then, has a ho seshoe in In (I) . Lemma 3 .2 . Le be a con inuous map o he in e al I = [a, a + 1] in o i sel such ha (a+1) = (a)+d wi h dE {-1,0, 1} . Assume ha w x ( ) is a simple se o any xE I and ha a E w x ( ) and a + 1 E w y ( ) o some x, y EI . Then he ollowing s a emen s hold : (a) I w x( ) 7~ w y ( ) hen w x ( ) and w y ( ) a e ixed poin s o . (b) I w x ( ) = w y ( ) hen w x ( ) is a pe iodic o bi o pe iod 2 o . P oo . I will be di ided in o se e al cases . Case 1 : d = 0 . We shall p o e ha {a, a + 1} ~ w( ) and hus he lemma holds . In iew o Theo em A and he ac ha o an in e al map he chain ecu en se is a union o simple se s i and only i each omega limi se is a simple se , i is enough o show ha i {a, a + 1} C w( ) hen has a ho seshoe . I (a+ 1) = (a) = a hen, since a+ 1 E w y ( ) and w y ( ) is in a ian , i ollows ha he e exis s z E I such ha (z) = a+ 1 . Thus, [a, z] and [z, a + 1] o m a ho seshoe o . 312  LL . ALSEDÁ, V . FEDORENKO I (a + 1) = (a) = a + 1 we p oceed in a simila way o ob ain a ho seshoe o . Assume now ha a < (a) = (a + 1) < a + 1 . As be o e, he e exis z, z' É I such ha (z) = a and (z') = a + 1 . Mo eo e , he e exis s , a ixed poin o be ween z and z' . I z < z' and (a) >_ hen he in e als [a, z] and [z, ] o m a ho se- shoe o . I z < z' and (a) < hen he ho seshoe o is gi en by [ , z'] and [z', a + 1] . Thus, i z < z' we a e done . I z' < z and (a) > we se A = [a, z'], B= [z', ] and C= [ , z] . Then we ha e (A) D C, (B) D C and (C) D A U B . Hence, 2 (A) D AUB  and  2 (B) D AUB . Thus A and B o m a ho seshoe o . I z' < z bu (a) < hen, in a simila way we can see ha [ , z] and [z, a + 1] o m a ho seshoe o . Case 2 : d = 1 . We ha e ha (a) = a and (a + 1) = a + 1 . Thus, w ., ( ) and w y ( ) a e simple se s which con ain a ixed poin . In iew o Rema k 1 .1 we ob ain ha w, : ( ) = {a} and w y ( ) = {a + 1} . Case 3 : d = -1 . We ha e ha {a, a + 1} is now a pe iodic o bi o o pe iod 2 . Thus, again by Rema k 1 .1 we see ha w ., ( ) = w y ( ) _ {a, a + 1} . Le now bea ci cle map, and le F be a li ing o . We say ha E A i he e exis x E lié and p E 7L such ha (F - p) ([x, x + 1]) C [x, x + 1] . Tha is, i F sa is ies (b) o Lemma 2 .1 wi h q = 1 . F om Rema k 2 .2 we no e ha deg( ) E {-1,0,1} . Lemma 3 .3 . Each map E A has oneand only one om he ollow- ing wo p ope ies : (a) has a ho seshoe . (b) wx( ) is a simple se o each xES 1 . P oo :: Le F be a li ing o . Since E A he e exis x E I[8 and pE 9L such ha (F - p) ([x, x + 1]) C [x, x + 1] . In iew o Theo em A, Lemmms 3 .1 and 3 .2 and he ac ha he chain ecu en se o an in e al map is union o simple se s i and only i each omega limi se is a simple se we ge ha (F - p) has one and only one om he ollowing wo p ope ies . (i)  (F - p) has a ho seshoe in (x, x + 1) . (ii) w y (F-p) is a simple se o each y E [x, x+1] and i x Ew xl (F-p) and x+1 E W X2 (F-p) o some X1, X2 E [x, x+1] hen w xl (F-p) SIMPLE AND COMPLEX DYNAMICS FOR CIRCLE MAPS  31 3 and w ., Z ( - p) a e ixed poin s o (F - p) when w ., l (F - p) = w~, 2 (F - p) and w, : l (F - p) is a pe iodic o bi o pe iod 2 when w ., (F - p) :~ w-2 (F - p) . Since (F - p) is also a li ing o and e : (x, x + 1) -> S 1 {e (x)} is a homeomo phism we ge he desi ed conclusion . Now we a e eady o p o e he main esul o his pape . P oo o Theo em 1 .2 : I ollows s aigh o wa dly om Lemmas 2 .1 and 3 .3 and om he ac ha i sa is ies (a) o (b) o Lemma 2 .1 hen Pe ( ) :?É 0 . 4 . A geome ic iew-poin o Condi ion (b) o Theo em 1 .2 . Th oughou his sec ion we will assume ha is a ci cle map sa is ying Condi ion (b) o Theo em 1 .2 . We shall y o explain his Condi ion in e ms o he g aph o he map and he chain ecu en se o . FYom Rema k 2 .2 we ha e ha deg( ) E {-1, 0,1} .  I deg( ) E {-1, 0}, by looking a he p oo o Theo em 1 .2, we can p o e ha he si ua ion o he chain ecu en se o is he same as in he case o in e al maps . Tha is, he chain ecu en se o is he union o all simple se s . Now assume ha deg( ) = 1 . Also om he p oo o Lemma 2 .1 and om he ac ha a ci cle map sa is ies one and only one om he condi ions o Theo em 1 .2 we see ha i F is a li ing o hen LF = {p1 q} wi h q E N and p E 7L ela i ely p ime . By using he no a ion om he p oo o Lemma 2 .1 we see ha he e exis s a pe iodic (mod . 1) poin 0 o F wi h o a ion numbe p/q such ha I we se (Fq _p)([0,0+11) C [0 , 0 + 11 . Q={FZ(~3)+m : i=0,1,2,  ,q-1,mEZ}= = e  1({ z(e(a)) :  i = 0, 1, 2,  . . , q - 1}) _ _ { . . . ~3-2, 0-1, /30, Q1, 02 . . . } hen i is no di icul o p o e ha (Fq -p)([~i,Qi+1]) C [~3i,~3i+1] o all i E 7G and ha o each i, j E 7G he diag am