ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES

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1 ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES KARMA DAJANI AND CHARLENE KALLE Abstract. Let T be a ergodic trasformatio o X ad {α } a sequece of partitios o X. Defie K (x) = mi{j 1 : T j x α (x)}, where α (x) is the elemet of α cotaiig x. I this paper we give coditios o T ad α, for which (x) exists. We study the questio i oe ad higher dimesios. 1. Itroductio Let x [0, 1) ad let T be a trasformatio from [0, 1) to [0, 1). Oe ca study the movemet of x uder iteratios of T, ad if B is a set of o-egative measure cotaiig x, oe ca address the questio of how may iteratios of T are eeded for the orbit of x to retur to the set B for the first time. This umber of iteratios is called the first retur time of x to B uder T. The first ad most famous result about this type of questios is the Poicaré Recurrece Theorem. It states that if B is a set of positive measure ad if T is a trasformatio that preserves a fiite measure, the almost all elemets of B retur to B evetually. A immediate cosequece of this result is that almost all x B retur to B ifiitely ofte. The Poicaré Recurrece Theorem is powerful, but it does t make a statemet about the speed with which a poit returs to a set it started i. A great umber of articles have bee published aswerig this questio i various ways. Usually the first retur time of x to a partitio elemet geerated by the trasformatio T itself is studied. Give a partitio P of [0, 1) ad a trasformatio T, it is possible to costruct a sequece of partitios {P } by lookig at iverse images of P uder iteratios of T ad takig itersectios. More precisely, P = 1 i=0 T i P = {A 0 T 1 A 1 T ( 1) A 1 : A i P}. Usig the partitio P ad the trasformatio T, oe ca costruct for almost every x [0, 1) a expasio. This is doe by specifyig for each the partitio elemet of P that the elemet T x belogs to. The expasio that is obtaied i this way is called the (T, P)-expasio of x. If P (x) deotes the partitio elemet of P that cotais x, the P (x) specifies the first coordiates of the (T, P)-expasio of x. Orstei ad Weiss proved a asymptotic result o the umber of iteratios of T that are eeded before x returs to P (x) for the first time. Let (X, F, µ, T ) be 1991 Mathematics Subject Classificatio. Primary 28D05, 37A35. Key words ad phrases. first retur times, Shao-McMilla-Breima Theorem, Poicaré recurrece. 1

2 2 KARMA DAJANI AND CHARLENE KALLE a dyamical system ad let E X. The the first retur time of a elemet x of E to the set E will be deoted by R E (x) ad is give by R E (x) = mi{j 1 : T j x E}. The the followig theorem states the result of Orstei ad Weiss. For the proof, cosult [O]. Theorem 1.1. Let (X, F, µ, T ) be a ergodic ad measure preservig dyamical system with µ a probability measure ad suppose P is a partitio o X with H(P) <. Defie the sequece of partitios {P } by P = 1 i=0 T i P for each 1. The log R P(x)(x) = h(t, P) µ a.e. I other words, if we look at the (T, P)-expasio of x, the above theorem says that the first coordiates of this expasio will repeat themselves for the first time after approximately 2 h steps. Others proved similar results for more specific trasformatios or more geeral sets. Marto ad Shields for example studied the asymptotic covergece of logarithmic waitig times. The waitig time of a elemet y (ot ecessarily i E) to E uder T is deoted by W E (y) ad is give by W E (y) = mi{j 1 : T j 1 y E}. Marto ad Shield showed that also i this case we have to wait 2 h(t,p) steps, before we see the first coordiates agai for the first time. Theorem 1.2. Let (X, F, µ) be a probability space ad T a measure preservig ad ergodic trasformatio o X that is weak Beroulli. Suppose that P is a partitio o X with H(P) <. Let x, y X with P(x) P(y) ad defie agai the sequece of partitios {P } by P = 1 i=0 T i P for each 1. The log W P(x)(y) = h(t, P) µ a.e. A proof of this theorem ca be foud i [M]. Seo studied the first retur time of a poit uder a specific trasformatio, amely the irratioal traslatio, to a dyadic iterval cetered at the poit itself. The proof ca be foud i [S]. Recall that a irratioal umber θ (0, 1) is of type η if η = sup{β : if j jβ jθ = 0}, where θ equals the distace to the earest iteger. Theorem 1.3. Let θ (0, 1) be a irratioal umber. Let furthermore (I, B, λ, T ) be a dyamical system, with I = [0, 1), B the Borel σ-algebra o I, λ the Lebesgue measure o (I, B) ad T the trasformatio o I defied by T x = x + θ (mod 1). The θ is of type 1 if ad oly if log R B(x,2 )(x) = 1, where B(x, 2 ) = {y [0, 1) : y x < 2. I the proof of this theorem it is show that sup log R B(x,2 )(x) = 1 ad if log R B(x,2 )(x) = 1 η,

3 ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES 3 where η deotes the type of the irratioal umber θ. A more geeral versio of the theorem above is give by Barreira ad Saussol. Let X R d for some d N. For each x X ad r, ɛ > 0 we ca defie the set A ɛ (x, r) = {y B(x, r) : R B(x,r) (y) µ(b(x, r)) 1+ɛ }, where B(x, r) is the ball i R d with cetre x ad radius r. A measure µ is said to have log retur time with respect to a trasformatio T if if log µ(a ɛ (x, r)) log µ(b(x, r)) > 1. Barreira ad Saussol proved the ext theorem. The proof as well as examples of trasformatios havig the above property ca be foud i [B]. Theorem 1.4. Let T : X X be a Borel measurable trasformatio o a measurable set X R d for some d N ad µ a T -ivariat probability measure o X. If µ has log retur time with respect to T ad if for µ-almost every x X, the if if log R B(x,r) (x) log r ad log R B(x,r) (x) sup log r for µ-almost every x X. log µ(b(x, r)) log r = if = sup > 0 log µ(b(x, r)) log r log µ(b(x, r)) log r I this paper we will ot be lookig at retur times to sets geerated by the trasformatio itself, but at retur times to sets geerated by aother trasformatio. Let S be a trasformatio ad α a partitio, together geeratig the sequece of partitios {α }, where α = 1 i=0 S i α. We ca the look at the (S, α)-expasio of a elemet x ad if α (x) deotes the partitio elemet of α cotaiig x, the this partitio elemet specifies the first coordiates of this expasio. If T is aother trasformatio defied o the same space ad if we defie the value K (x) by K (x) = mi{j 1 : T j x α (x)}, the K (x) deotes the first retur time of x uder T to the partitio elemet of α it started i. I other words, K (x) is the first j, such that the first coordiates of the (S, α)-expasio of T j x equal those of the (S, α)-expasio of x itself. A importat differece of this setup with respect to the results metioed above, is that the partitios uder cosideratio are idepedet of the trasformatio T ad are i fact geerated by the other trasformatio S. We will show that also the logarithm of the quatity K (x) coverges asymptotically, but surprisigly this it does t deped o the trasformatio T at all. We will prove that for almost all elemets x, (x) = h, where h is the etropy of S relative to the partitio α. This meas that the covergece of this value depeds oly o the radomess of S ad ot o that

4 4 KARMA DAJANI AND CHARLENE KALLE of T. Thus, the same result holds for all trasformatios T satisfyig the right coditios. This geeralizes a theorem of Kim ad Kim [K], where they proved the result for the case {α } is the collectio of dyadic itervals of order. We will begi by statig ad provig the oe-dimesioal versio of the result metioed above. I the last sectio we will give geeralizatios of this result to higher dimesios by usig the Theorem of Barreira ad Saussol. This is doe for specific choices of S ad α. Throughout this text, if α is a partitio of a o-empty set X ad x is a elemet of X, the α(x) will deote the partitio elemet of α cotaiig x. 2. Oe-dimesioal Log Retur Times I this sectio we will study the asymptotic behaviour of the log retur time for certai oe-dimesioal partitios defied o the uit iterval I = [0, 1). Therefore, cosider the dyamical systems (I, B, µ 1, T ) ad (I, B, µ 2, S), where I is the uit iterval [0, 1), B is the Borel σ-algebra o I, ad µ 1 ad µ 2 are probability measures o (I, B) with µ 1 µ 2. Suppose T ad S are ergodic trasformatios from I to itself ad µ 1 -, respectively µ 2 -ivariat. Let α be a iterval partitio of I, with H(α) < ad such that each partitio α = 1 i=0 S i α is agai a iterval partitio of I. Throughout this sectio, we let h = h µ2 (S, α) idicate the etropy of S with respect to α, ad we suppose h > 0. Defie for every x [0, 1) the first retur time of x uder T to its cylider set α (x) by K (x) = mi{j 1 : T j x α (x)}. Theorem 2.1. Let P be a fiite partitio of I, cosistig of itervals. Defie the sequece of partitios {P } 1 by P = 1 i=0 T i P ad suppose that for each 1 the partitio P also cosists of itervals oly. Suppose furthermore that H(P) < ad h µ1 (T, P) > 0. The (x) = h µ 1 a.e. Remark. Notice that to guaratee that each of the partitios P is a iterval partitio, it is eough that the trasformatio T is mootoe o each of the elemets of the partitio P. Proof. Let h = h µ1 (T, P), ad let 0 < ɛ < h be give. Call a elemet A of (h +ɛ) P (, ɛ)-typical if 2 < µ 1 (A) < 2 (h ɛ), ad a similar defiitio for members of α. Let l (P) be the umber of (, ɛ)-typical sets for the partitio P, ad l (α) be the umber of (, ɛ)-typical sets for the partitio α. By the Shao- McMilla-Breima Theorem, P (x) ad α (x) are (, ɛ)-typical for all sufficietly large, ad for almost every x,. Furthermore, there exist a positive iteger N such that l (P) (ɛ) < 2 (h +ɛ) ad l (α) (ɛ) < 2 (h+ɛ) for all > N. For each > N, defie the iteger m = m() = (h ɛ) h +2ɛ ad let 2 m(h +ɛ) < µ 1 (P m (x)) < 2 m(h ɛ) D (ɛ) = x I : 2 (h+ɛ) < µ 2 (α (x)) < 2 (h ɛ). α (x) P m (x) If x D (ɛ), the α (x) will itersect at least two elemets of the partitio P m ad the elemets with which it itersects iclude at least oe (m, ɛ)-typical elemet

5 ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES 5 of P m, amely P m (x). Sice both partitios P m ad α cosist of itervals oly, the measure of D (ɛ) ca be estimated by µ 2 (D (ɛ)) 2 the umber of (m, ɛ)-typical elemets of P m the maximal µ 2 -measure of a (, ɛ)-typical elemet of α 2 2 m(h +ɛ) 2 (h ɛ) 2 2 ((h ɛ) (h ɛ) h +ɛ h +2ɛ ). By the Borel-Catelli Lemma this meas that µ 2 (D (ɛ) i.o.) = 0 ad the Shao- McMilla-Breima Theorem gives that µ 2 ({x I : α (x) P m (x) i.o.}) = 0. Sice µ 1 µ 2, it follows that µ 1 ({x I : α (x) P m (x) i.o.}) = 0, so for almost all x I, for big eough, α (x) P m (x). Therefore K (x) R Pm(x)(x) ad by Theorem 1.1 the if (x) log R Pm(x)(x) if m m h = (h ɛ) h +2ɛ µ 1 a.e. Sice this holds for all ɛ sufficietly small, we have that if (x) h µ 1 a.e. The fact that sup (x) h for µ 1 -almost every x ca be prove i a similar way by takig ɛ < h, replacig the set D (ɛ) by 2 m(h +ɛ) < µ 1 (P m (x)) < 2 m(h ɛ) D (ɛ) = x I : 2 (h+ɛ) < µ 2 (α (x)) < 2 (h ɛ), P m (x) α (x) ad takig m = m() = (h+2ɛ) h ɛ. Notice that i this case we ca immedeately give a estimate of the µ 1 -measure of D (ɛ), so that we do ot have to impose that µ 2 µ 1. Now cosider the umber K (x, y) = mi{j 1 : T j 1 y α (x)}, which ca be iterpreted as the time we have to wait util a elemet y X eters for the first time the partitio elemet of α i which x lies. We are goig to prove the equivalece of Theorem 2.1 for this waitig time. Theorem 2.2. Let P be a fiite partitio of I, cosistig of itervals ad such that each of the partitios P = 1 i=0 T i P is agai a iterval partitio. Suppose furthermore that H(P) < ad h µ1 (T, P) > 0. If T is a measure preservig, ergodic, weakly Beroulli trasformatio, the (x, y) = h (µ 1 µ 1 ) a.e. Proof. Fix y I, ad let ɛ > 0. Defie 2 (h+ɛ) < µ 2 (α (x)) < 2 (h ɛ) A = x I : K (x, y) < 2 (h 2ɛ) The A is the uio of those (, ɛ)-typical elemets of α, that cotai T j 1 y for some j < 2 (h 2ɛ). Sice this umber caot exceed 2 (h 2ɛ), we have µ 2 (A ) < 2 (h 2ɛ) 2 (h ɛ) = 2 ɛ..

6 6 KARMA DAJANI AND CHARLENE KALLE The by the Borel-Catelli Lemma, the Shao-McMilla-Breima Theorem ad the fact that µ 1 µ 2, we have, as i the proof of Theorem 2.1, that if (x, y) if log 2 (h 2ɛ) = h 2ɛ (µ 1 µ 1 ) a.e. To prove that sup (x,y) h for (µ 1 µ 1 )-almost every (x, y), we let h = h µ1 (T, P), where 0 < ɛ < h is give. For each defie the umber m = m() = (h+2ɛ ) h ɛ. As i the proof of Theorem 2.1 we ca show that for big eough, we have for almost all x I that P m (x) α (x). This meas that the K (x, y) W m (x, y) ad therefore by Theorem 1.2 sup (x, y) log W m (x, y) sup m m = (h + 2ɛ h ) h ɛ (µ 1 µ 1 ) a.e. Sice this holds for all 0 < ɛ < h, we have the result. I the previous theorems, the coditio that h(t, α) > 0 was used i the proof. We will ow show that this coditio is ideed ecessary by showig that for certai irratioal rotatios K (x)/ does ot exist. To this ed, cosider the probability space (I, B, λ) ad let the trasformatio T θ : I I be defied by T θ x = x + θ (mod 1) with θ (0, 1) a irratioal umber. It is well kow that h λ (T, P) = 0 for ay partitio P. Suppose α is a geeratig partitio of S with H(α) < ad h = h(s, α) > 0. Defie for each x I, We have the followig theorem. K θ (x) = mi{j 1 : T j θ x α (x)}. Theorem 2.3. Let θ (0, 1) be a irratioal umber of type η. The, for almost all x I if (x) θ = h ad sup (x) θ = h a.e. η Proof. Let 0 < ɛ < h be give. The by the Shao-McMilla-Breima Theorem, we kow that for almost all x, there exists a N such that for all N we have Therefore, for these x s we have that 2 (h+ɛ) < λ(α (x)) < 2 (h ɛ). α (x) B(x, 2 (h ɛ) ), so K (x) R B(x,2 )(x). O the other had, if we defie for each N, the (h ɛ) iteger m = m() = (h + 2ɛ) ad the set 2 (h+ɛ) < λ(α (x)) < 2 (h ɛ) E (ɛ) = x I : B(x, 2 m, ) α (x) by a argumet similar to the proof of Theorem 2.1 we ca see that =1 λ(e (ɛ)) <. Thus by the Borel Catelli Lemma ad the Shao-McMilla-Breima Theorem K θ (x)(x) R B(x,2 m )(x) for all sufficietly large. By Theorem 1.3 if θ (x) = h η λ a.e.,

7 ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES 7 ad sup θ (x) = h a.e. The followig corollary follows immediately from the above theorem. Corollary 2.1. Let θ (0, 1) be a irratioal umber of type η. If η = 1, the (x) θ = h a.e., ad if η > 1, the (x) θ does ot exist. 3. Higher Dimesios Usig the Theorem of Barreira ad Saussol, some of the results of the previous sectio ca be exteded to dimesio d N i case the partitios α are give by the d-dimesioal dyadic hypercubes of order. To this ed, cosider the probability space (I d, B d, λ d ), where I d is the d-dimesioal uit hypercube, B d is the d-dimesioal Borel σ-algebra ad λ d is the d-dimesioal Lebesgue measure o (I d, B d ). Let the partitios Q d of I d be give by The for each x I d, d 1 Q d = { [ i j 2, i j ) : 0 i 0,... i d 1 2 1}. j=0 (diameter Q d ( x)) = ( ad λ d (Q d ( x)) = 2 d. Let Q (i1,...,i d) d (2 ) 2 ) 1/2 = d 2 i=1 deote the partitio elemet Q (i1,...,i d) = [ i 1 2, i )... [ i d 2, i d ). Usig the result of Barreira ad Saussol, we prove the followig theorem. Theorem 3.1. Let T be a measure preservig trasformatio o the probability space (I d, B d, µ). Assume that µ has log retur time with respect to T, ad is absolutely cotiuous with respect to λ d with desity g bouded away from zero ad bouded from above. Defie for each x I d, the for µ-almost every x I d, K d ( x) = mi{k N : T k x Q d ( x)}, ( x) d = d. log λ d (B( x, r)) Proof. Notice first that = d. Sice µ λ d with desity log r bouded away from zero ad bouded from above, it follows that log µ(b( x, r)) log λ d (B( x, r)) = = d. log r log r

8 8 KARMA DAJANI AND CHARLENE KALLE Therefore all the coditios of the Theorem of Barreira ad Saussol are satisfied. Sice for each x I d, Q d ( x) B( x, d 2 ), we have K( x) d R B( x, d 2 ) ( x) ad thus if d ( x) Usig Theorem 1.4 the gives if if if d ( x) log R B( x, d 2 ) ( x) log( log( d 2 ) d 2 ) log R B( x,r) ( x) (1 r log d). log r if log µ(b( x, r)) log r = d for µ-almost all x I d. For the secod part, usig Kac s Lemma we get 2 1 K( x)dµ( x) d = K( x)dµ( x) d 2 d. I d i 1,...,i d =0 Q (i 1,...,i d ) Let ɛ > 0. For each 1, defie the set B d = { x I d : K( x) d > 2 d(1+ɛ) }. The by Markov s Iequality µ(b) d 1 K 2 ( x)dµ( x) d 2 dɛ. d(1+ɛ) I d By the Borel-Catelli Lemma, the µ(b d i.o.) = 0, so that sup d ( x) sup log 2 d(1+ɛ) = d + dɛ µ a.e. The above theorem ca be geeralized whe the partitios Q d are replaced by d-fold product partitios. α α, where α is some iterval partitio of [0, 1). For ease of otatio, we restrict our attetio to dimesio 2. Give a rectagle R, we defie the frame of R of width δ as the set F(R, δ) = { x I 2 : x R ad d( x, R) δ}, where R is the boudary of R ad d idicates the usual Euclidia distace i the plae. Notice that the proportio of R take up by its frame of width δ is bouded above by 4 (legth of R) δ. λ(r) Now, cosider the probability space (I, B, ν). Suppose S is a measure preservig weakly mixig trasformatio o I, ad let α be a iterval partitio o I with H λ (α) <. Costruct the sequece of partitios {α } by settig α = 1 i=0 S i α, ad suppose that h = h(s, α) > 0. Assume that for each 1 the partitio α cosists of itervals oly. Now costruct the two-dimesioal dyamical system (I 2, B, ν, S S) by settig I 2 = I I, B = B B ad ν = (ν ν) The h = h(s S, α α) = 2h. We assume throughout that ν is equivalet to λ with desity bouded away from 0 ad. This allows us to replace ν by λ i the Shao- McMilla-Breima theorem applied to the partitio α. For this reaso, we shall assume with o loss of geerality that ν is λ. Let T be a trasformatio o (I 2, B, µ), where µ is a T -ivariat probability measure, absolutely cotiuous with respect to λ with desity bouded away from

9 ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES 9 zero ad bouded from above. Assume furthermore that T has log retur time with respect to µ. Defie K ( x) = mi{j 1 : T j x (α α )( x)}. The also i this case we have the followig theorem. Theorem 3.2. For µ-almost every x I 2, log K ( x) = h. log Proof. Agai the proof is doe by first checkig that if K ( x) h µ log a.e. ad the that sup K ( x) h for µ-almost all x. For the first part, let ɛ > 0 be give ad defie for each 1 2 (h+ɛ) < λ(α (x)) < 2 (h ɛ) D (ɛ) = x = (x, y) I 2 : 2 (h+ɛ) < λ(α (y)) < 2 (h ɛ). The for all x D (ɛ), we have that (α (x) α (y)) B( x, 2 2 (h ɛ) ), so R B( x, 2 2 (h ɛ) ) ( x) K ( x). The Shao-McMilla-Breima Theorem tells us that λ(d (ɛ) c ) = 0, so by absolute cotiuity also µ(d (ɛ) c ) = 0 ad the by Theorem 1.4 if log K ( x) log R B( x, 2 2 if (h ɛ) ) ( x) log( 2 2 (h ɛ) ) h 2ɛ µ a.e. log( 2 2 (h ɛ) ) For the other part, let ɛ > 0 agai be give ad defie for each 1 the umber m = m() = (h + 4ɛ ) ad the set 2 (h+ɛ ) < λ(α (x)) < 2 (h ɛ ) D (ɛ ) = x = (x, y) I : 2 (h+ɛ ) < λ(α (y)) < 2 (h ɛ ) B( x, 2 m ) (α (x) α (y)). If x D (ɛ ), the B( x, 2 m ) overlaps with at least two elemets of the partitio α α, so that x = (x, y) must lie i the frame of α (x) α (y) of width at most 2 2 m. Therefore, λ(f(α (x) α (y), 2 m )) λ(α (x) α (y)) The K ( x) R B( x,2 m )( x), so by Theorem 1.4 sup log K ( x) sup log R B( x,2 m )( x) log 2 m 2 (h ɛ ) 2 2 m 2 ( h+2ɛ ) Ad sice this holds for all ɛ > 0, this fiishes the proof. 2 2 ɛ. m 2(h + 4ɛ ) = h + 8ɛ µ a.e. A similar result ca be obtaied without usig the Theorem of Barreira ad Saussol, but a differet set of assumptios istead. Let us cosider a dyamical system (I 2, B, ν, S), where ν λ with desity fuctio f bouded away from zero by a costat c S > 0 ad from above by a costat C S <. Let S be a ergodic ad ν-preservig trasformatio. Also cosider a fiite partitio α of I 2 with H ν (α) < ad h = h ν (S, α) > 0. Suppose furthermore that for each 1, the

10 10 KARMA DAJANI AND CHARLENE KALLE partitio α = 1 i=0 S i α cosists of rectagles ad that there exists a costat ζ α > 1 idepedet of such that for all A α, (diameter A) 2 ζ α λ(a). Let (I 2, B, µ, T ) also be a dyamical system where µ λ, with desity g bouded away from zero by a costat c T > 0 ad bouded from above by a costat C T <. Suppose furtermore that T is a ergodic ad µ-preservig trasformatio. We the have the followig theorem. Theorem 3.3. Let P be a fiite partitio of I 2 with H µ (P) <. Suppose furthermore that h = h µ (T, P) > 0. Defie the sequece of partitios {P } by P = 1 i=0 T i P ad suppose that for all 1 the partitio P cosists of rectagles oly. Suppose furthermore that there exists a costat ζ P, such that for all 1 (diameter P ) 2 ζ P λ(p ) for all P P. The log K ( x) = h µ a.e. Notice that istead of the assumig a log retur time, we ow imposed the coditio that the partitio elemets cosist of rectagles, whose diameters ca ot become large with respect to their measures. Proof. Let ɛ > 0 be give. For each 1, defie the umber m = m() = (h 2ɛ) h +ɛ ad the set 2 m(h +ɛ) < µ(p m ( x)) < 2 m(h ɛ) D (ɛ) = x I 2 : 2 (h+ɛ) < ν(α ( x)) < 2 (h ɛ). α ( x) P m ( x) Usig the same techique as i the secod part of the proof of Theorem 3.2, we ca show that λ(f(p m ( x), δ)) ζp ζ α C T 4 2 λ(p 1/2ɛ. m ( x)) c S From this we ca deduce that λ(d (ɛ)) <. =1 So by usig the Borel Catelli Lemma, it ca be show that for big eough α ( x) P m ( x), which meas that ( x) if if The proof of the assertio that sup log R Pm( x)( x) m ( x) h m h = (h 2ɛ) h + ɛ µ a.e. ca be obtaied i a similar way by takig m = m() = (h+2ɛ) h ɛ ad 2 m(h +ɛ) < µ(p m ( x)) < 2 m(h ɛ) D (ɛ) = x I 2 : 2 (h+ɛ) < ν(α ( x)) < 2 (h ɛ). P m ( x) α ( x) µ a.e.

11 ON THE CONVERGENCE OF LOGARITHMIC FIRST RETURN TIMES 11 Refereces [B] Barreira, L. ad Saussol B. Hausdorff Dimesios of Measures via Poicaré Recurrece i Commuicatios i Mathematical Physics. pp : Spriger-Verlag. [D] [K] [M] [O] [S] [S1] Dajai, K., De Vries, M. ad Johso, A. The Relative Growth of Iformatio i Two-Dimesioal Partitios Kim, C. ad Kim, D.H. O the Law of Logarithm of the Recurrece Time i Discrete ad Cotiuous Dyamical Systems, Volume 10, Number 3, 2004, pp Marto, K. ad Shields, P. Almost-sure Waitig Time Results for Weak ad Very Weak Beroulli Processes i Ergodic Theory of Dyamical Systems, Volume 15, 1995, pp Orstei, D.S. ad Weiss, B. Etropy ad Data Compressio Schemes i IEEE Trasactios o Iformatio Theory, Volume 39, Number 1, 1993, pp Choe, G. H.; Seo, B. K. Recurrece speed of multiples of a irratioal umber. Proc. Japa Acad. Ser. A Math. Sci. 77 (2001), o. 7, Seo, B. K. Doctoral Thesis: Recurrece of Multiples of Irratioal Numbers Modulo , Korea Advaced Istitute of Sciece ad Techology, Departmet of Mathematics. Fac. Wiskude e Iformatica, Uiversiteit Utrecht, Postbus , 3508 TA Utrecht, the Netherlads address: dajai@math.uu.l Fac. Wiskude e Iformatica, Uiversiteit Utrecht, Postbus , 3508 TA Utrecht, the Netherlads address: kalle@math.uu.l