Full shifts and irregular sets

Transcription

1 São Paulo Joural of Mathematical Scieces 6, 2 (2012), Full shifts ad irregular sets Luis Barreira Departameto de Matemática, Istituto Superior Técico Lisboa, Portugal address: barreira@math.ist.utl.pt Jiju Li Departameto de Matemática, Istituto Superior Técico Lisboa, Portugal Departmet of Mathematics, Zhagzhou Normal Uiversity Zhagzhou, , P. R. Chia address: li-jiju@163.com Claudia Valls Departameto de Matemática, Istituto Superior Técico Lisboa, Portugal address: cvalls@math.ist.utl.pt Dedicated to Luís Magalhães ad Carlos Rocha o the occasio of their 60th birthdays Abstract. By Birkhoff s ergodic theorem, the set of poits X ϕ for which the Birkhoff averages of a cotiuous fuctio ϕ diverge has zero measure with respect to ay fiite ivariat measure. Thus, at least from the poit of view of ergodic theory, this set could ot be smaller. Nevertheless, it ca be large from other poits of view. For example, for subshifts with the weak specificatio property, we showed recetly that X ϕ is residual wheever it is oempty (it is a simple exercise to show that X ϕ is dese wheever it is oempty). The mai purpose of this ote is to covey i the simplest possible maer the proof of our result i the particular case of the full shift o a fiite umber of symbols. This has the advatage of avoidig some accessory techicalities that are ecessary i the geeral case. I fact, we cosider also the more geeral case whe the set of accumulatio poits of the Birkhoff averages of a cotiuous fuctio is a prescribed closed iterval ad we show that it is residual wheever it is oempty Mathematics Subject Classificatio. Primary 37B10. Key words: Full shifts, irregular sets, residual sets. 135

2 136 L. Barreira, J. Li, ad C. Valls 1. Itroductio We first itroduce the otio of the irregular set for the Birkhoff averages of a give fuctio. Give a cotiuous map f : X X o a compact metric space, the irregular set for the Birkhoff averages of a fuctio ϕ: X R is defied by X ϕ = { 1 1 x X : lim if ϕ(f i (x)) < lim sup 1 1 } ϕ(f i (x)). As a cosequece of Birkhoff s ergodic theorem, the irregular set has zero measure with respect to ay fiite f-ivariat measure µ o X (this meas that µ(f 1 A) = µ(a) for ay measurable set A X). Theorem 1. For a cotiuous map f o a compact metric space, if the fuctio ϕ is cotiuous, the µ(x ϕ ) = 0 for ay f-ivariat fiite measure µ o X. O the other had, it was show i [4] that from the poit of view of topological dyamics ad dimesio theory the set X ϕ ca be as large as the whole space. We formulate oly a particular case of the results, for the full shift, which is also the dyamical system cosidered i this ote. Theorem 2. For the full shift f o a fiite umber of symbols, if the fuctio ϕ is Hölder cotiuous, the X ϕ is either empty or has Hausdorff dimesio equal to the Hausdorff dimesio of the whole space X. This pheomeo was first observed by Pesi ad Pitskel i [9] for the full shift o two symbols. We refer the reader to the book [1] for a detailed discussio ad to [3, 5, 6, 7, 8, 10] for related work. Besides the Hausdorff dimesio oe may also cosider the topological etropy ad more geerally the topological pressure of the irregular set. Here we cosider yet aother poit of view for which a irregular set ca be very large, ulike what happes from the poit of view of ergodic theory. Namely, for the full shift o a fiite umber of symbols ad for a arbitrary cotiuous fuctio ϕ, we show that the set X ϕ is either empty or residual (we recall that a set is said to be residual if it cotais a dese G δ set). Theorem 3. For the full shift f o a fiite umber of symbols, if the fuctio ϕ is cotiuous, the X ϕ is either empty or residual. This is a particular case of results of ours i [2] that cosider the geeral class of subshifts with the weak specificatio property. Roughly speakig, a symbolic system is said to have the weak specificatio property if uder

3 Full shifts ad irregular sets 137 iteratio oe ca go from oe cylider set to aother (see (1) for the defiitio) evetually stayig outside both of them for a bouded period of time, idepedetly of the iitial ad fial cylider sets. I fact, we show i the remaider of the paper that some subsets of the irregular set are also residual. Namely, give a iterval I R, let X I = { x X : A ϕ (x) = I }, where A ϕ (x) is the set of accumulatio poits of the sequece of Birkhoff averages 1 1 ϕ(f i (x)). This is the cotet of Theorem 4, of which Theorem 3 is a corollary. We refer the reader to [2] for details. I order to show that the set X I is residual we bridge together strigs of sufficietly large legth correspodig to Birkhoff averages with differet limits i the iterval I. Goig back ad forth betwee strigs correspodig to these limits oe ca esure that the resultig Birkhoff averages diverge ad thus their iitial poits belog to the irregular set. The argumets i [2] are also ispired i this idea, although sice we are cosiderig arbitrary subshifts with the weak specificatio property various techical complicatios arise that to some extet hide the mai idea of the proof. 2. Formulatio of the result Let σ be the shift map o Σ = {1,..., k} N, where k 2 is a iteger. Moreover, let ϕ: Σ R be a cotiuous fuctio ad cosider the level sets { } B ϕ (α) = ω Σ : lim S ϕ(ω, ) = α, where S ϕ (ω, ) = 1 1 ϕ(σ i (ω)). We also cosider the oempty closed iterval L ϕ = { α R : B ϕ (α) } ad the set A ϕ (ω) of accumulatio poits of the sequece S ϕ (ω, ). Theorem 4. Let ϕ: Σ R be a cotiuous fuctio. iterval I L ϕ that is ot a sigleto, if the set Σ ϕ,i := { ω Σ : A ϕ (ω) = I } is oempty, the it is residual. Give a closed

4 138 L. Barreira, J. Li, ad C. Valls Proof. We first itroduce some otatio. For N, let Σ = {1,..., k} ad Σ = N Σ. For each ω Σ, we write ω = ad where [ω] = {ρ Σ : ρ = ω}, (1) (ω 1 ) = (ω 1 ω ). Give W 1,..., W Σ ad ω Σ, we write ωw 1 W = { ωω 1 ω : ω i W i, 1 i } ad W = W 1 W whe W 1 = = W = W. We proceed with the proof of the theorem. For each α R, N ad ε > 0, write F (α,, ε) = { ω : ω Σ ad S ϕ (ω, ) α < ε }. Now let k N ad choose α k,1,..., α k,qk I such that ad I q k i=1 B ( α k,i, 1/k ) (2) α k,i+1 α k,i < 1 k for i = 0,..., q k 1, α k,qk α k+1,1 < 1 k. (3) Moreover, let ε 1 > ε 2 > be a sequece of positive umbers decreasig to zero ad let 1,1 < 1,2 < < 1,q1 < 2,1 < 2,2 < < 2,q2 < be a sequece of positive itegers such that F (α k,i, k,i, ε k ) for k N, 1 i q k. It follows from Birkhoff s ergodic theorem that this choice ca be made. Let Ω 0 N Σ. For each ω Ω 0, we choose itegers {N k,i } k N,i=1,...,qk (depedig o ω) such that: (i) N 1,i 2 1,i+1 for 2 i q 1 1, N k,i 2 k,i+1 for k 2, 1 i q k 1, N k,qk 2 k+1,1 for k 1; (ii) N k,i+1 2 ω +N 1,1 1,1 +N 1,2 1,2 + +N k,i k,i, N k+1,1 2 ω +N 1,1 1,1 +N 1,2 1,2 + +N k,qk k,qk for k N, 1 i q k 1.

5 Full shifts ad irregular sets 139 Moreover, we defie sets Ω k,i Σ for k N ad i = 1,..., q k by Ω 1,1 = ωf (α 1,1, 1,1, ε 1 ) N1,1, ω Ω 0 Ω 1,2 = ηf (α 1,2, 1,2, ε 1 ) N1,2, η Ω 1,1 ad so o. Fially, let Ω 2,1 = E k,i = η Ω 1,q1 ηf (α 2,1, 2,1, ε 2 ) N2,1, ω Ω k,i [ω] ad E = q k k=1 i=1 E k,i. Clearly, E is a G δ set sice each cylider set [ω] is ope. Moreover, by costructio, each set E k,i is dese ad so it follows from Baire s theorem that E is also dese. It remais to show that E Σ ϕ,i (sice the Σ ϕ,i cotais the dese G δ set E ad hece is residual). We must prove that A ϕ (ω) = I for ω E. We recall that for each ω E, there exists ω 0 Ω 0 such that We first show that I A ϕ (ω). Give ω ω 0 F (α 1,1, 1,1, ε 1 ) N 1,1. (4) α I q k i=1 B ( α k,i, 1/k ), take i k {1,..., q k } such that α B ( α k,ik, 1/k ). For simplicity of the expositio, we assume that i k {1, q k }. Let s rk = ω 0 + q 1 j=1 where r k = (k, i k ). We will prove that It follows from (6) that N 1,j 1,j + + i k j=1 N k,j k,j, (5) S ϕ (ω, s rk ) α rk 0 whe k. (6) S ϕ (ω, s rk ) α S ϕ (ω, s rk ) α rk + α rk α < S ϕ (ω, s rk ) α rk + 1 k 0

6 140 L. Barreira, J. Li, ad C. Valls whe k. Therefore, α A ϕ (ω) ad I A ϕ (ω). I order to prove (6), write s rk = s rk + N rk rk. (7) Sice α rk ϕ := max ϕ(ω), ω Σ we have s rk 1 ϕ(σ i (ω)) s rk α rk es rk 1 s rk 1 ϕ(σ i (ω)) s rk α rk + ϕ(σ i (ω)) N rk rk α rk (8) i=es rk 2 s rk ϕ + N rk 1 q=0 rk 1 ϕ(σ j (σ esr k +qr k (ω))) rk α rk. j=0 Now we cosider the umbers v (ϕ) = sup { ϕ(ω) ϕ(ω ) : ω, ω Σ, ω = ω } ad V (ϕ) = v j (ϕ). j=1 By (4) ad the defiitio of the set F (α rk, rk, ε k ), oe ca choose sequeces ω 0,..., ω Nr k 1 Σ such that σ esr k +qr k (ω) rk = ω q rk (9) ad S ϕ (ω q, rk ) α rk < ε k (10) for q = 0,... N rk 1. Deotig the last absolute value i (8) by C q, it follows from (9) ad (10) that C q S ϕ (σ esr k +qr k (ω), rk ) S ϕ (ω q, rk ) + S ϕ (ω q, rk ) α rk rk V (11) rk + ε k rk for q = 0,... N rk 1. Together with (8) this implies that s rk 1 ϕ(σ i (ω)) s rk α rk 2 s r k ϕ + N rk (V rk (ϕ) + rk ε k ).

7 Full shifts ad irregular sets 141 Now we observe that it follows from coditio (ii) that s rk /s rk teds to zero whe k. Ideed, usig (5), (7) ad coditio (ii), we have s rk s rk 1 = N r k rk 2esrk rk s rk s rk ad thus, s rk / s rk + whe k. Moreover, it follows from the uiform cotiuity of ϕ that v (ϕ) 0 whe. Hece, V (ϕ)/ 0 whe ad N rk V rk (ϕ) V rk (ϕ) 0 whe k. rk s rk By the defiitio of s rk (see (7)), we have s rk > N rk rk ad N rk /s rk < 1/ rk. Therefore, S ϕ (ω, s rk ) α rk < 2 s r k ϕ s rk whe k, which completes the proof of (6). + V rk (ϕ) rk + ε k 0 Now we show that A ϕ (ω) I. For each positive iteger > ω 0 there exist k N, i k {1, 2,..., q k } ad 0 p < N k,ik +1 such that s rk + p tk < s rk + (p + 1) tk, (12) where t k = (k, i k + 1). Notice that k whe. We claim that S ϕ (ω, ) α rk 0 whe. (13) For simplicity of the otatio, i a similar maer to that i the former iclusio we assume that i k q k. If (13) holds, the it follows from (2) that dist ( S ϕ (ω, ), I ) S ϕ (ω, ) α rk + dist(α rk, I) 0 whe k. Sice I is closed, we coclude that A ϕ (ω) I. Now we establish property (13). We have 1 s rk 1 ϕ(σ i (ω)) α rk ϕ(σ i (ω)) s rk α rk s rk +p tk 1 + ϕ(σ i (ω)) p tk α rk i=s rk 1 + ϕ(σ i (ω)) ( s rk p tk )α rk i=s rk +p tk (14)

8 142 L. Barreira, J. Li, ad C. Valls ad we deote the last two absolute values i (14) respectively by G ad H. We shall estimate each of these terms. I a similar maer to that i (9) ad (10), oe ca choose ω 0,..., ω p 1 Σ such that σ sr k +qt k (ω) tk = ω q tk (15) ad S ϕ (ω q, tk ) α tk < ε k (16) for q = 0,... p 1. Proceedig as i (11), it follows from (3), (15) ad (16) that S ϕ (σ sr k +qt k (ω), tk ) α rk S ϕ (σ sr k +qt k (ω), tk ) α tk + α tk α rk for q = 0,..., p 1. Therefore, G tk p 1 V tk (ϕ) tk + ε k + 1 k S ϕ (σ sr k +qt k (ω), tk ) α rk q=0 Moreover, by (12), we have ( Vtk (ϕ) p + ε k + 1 k tk ). (17) H 2( s rk p tk ) ϕ 2 tk ϕ. (18) Collectig the estimates (17) ad (18), we obtai S ϕ (ω, ) α rk S ϕ (ω, s rk ) α rk + 2 t k ϕ + p(v tk (ϕ) + tk ε k ) + p (19) t k k. I a similar maer to that i the proof of (6), oe ca show that the first term i (19) teds to zero whe. Moreover, usig (12) ad coditio (i), we obtai 2 tk ϕ 2 t k ϕ s rk 2 t k ϕ N rk 0 (20) whe. O the other had, it follows from (12) that p tk k 1 k 0 (21) ad p(v tk (ϕ) + tk ε k ) V tk (ϕ) + tk ε k tk 0 (22)

9 Full shifts ad irregular sets 143 whe (sice k whe ). Hece, property (13) follows readily from (19), (20), (22) ad (21). This completes the proof of the theorem. Ackowledgmets Research partially supported by FCT (grat PTDC/MAT/117106/2010 ad CAMGSD). Jiju Li was also supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ) ad the Educatio Committee of Fujia Provice (Grat No. JA11173). Refereces 1. L. Barreira, Dimesio ad Recurrece i Hyperbolic Dyamics, Progress i Mathematics 272, Birkhäuser, L. Barreira, J. Li ad C. Valls, Irregular sets are residual, preprit. 3. L. Barreira, B. Saussol ad J. Schmelig, Distributio of frequecies of digits via multifractal aalysis, J. Number Theory 97 (2002), L. Barreira ad J. Schmelig, Sets of o-typical poits have full topological etropy ad full Hausdorff dimesio, Israel J. Math. 116 (2000), E. Che, T. Küpper ad L. Shu, Topological etropy for divergece poits, Ergod. Theory. Dyam. Systems 25 (2005), A.-H. Fa, D.-J. Feg ad J. Wu, Recurrece, dimesio ad etropy, J. Lodo Math. Soc. (2) 64 (2001), D.-J. Feg, K.-S. Lau ad J. Wu, Ergodic limits o the coformal repellers, Adv. Math. 169 (2002), J. Li ad M. Wu, Divergece poits i systems satisfyig the specificatio property, Discrete Coti. Dy. Syst. 33 (2013), Ya. Pesi ad B. Pitskel, Topological pressure ad variatioal priciple for ocompact sets, Fuctioal Aal. Appl. 18 (1984), D. Thompso, The irregular set for maps with the specificatio property has full topological pressure, Dy. Syst. 25 (2010),