FIRST RETURN TIMES: MULTIFRACTAL SPECTRA AND DIVERGENCE POINTS

. We provide a detailed study of the quantitative behavior of ﬁrst return times of points to small neighborhoods of themselves. Let K be a self-conformal set (satisfying a certain separation condition) and let S : K → K be the natural self-map induced by the shift. We study the quantitative behavior of the ﬁrst return time, of a point x to the ball B ( x,r ) as r tends to 0. For a function ϕ : (0 , ∞ ) → R , let A ( ϕ ( r )) denote the set of accumulation points of ϕ ( r ) as r (cid:5) 0. We show that the ﬁrst return time exponent, log τ B x,r x − log r , has an extremely complicated and surprisingly intricate structure: for any compact subinterval I of (0 , ∞ ), the set of points x such that for each t ∈ I there exists arbitrarily small r > 0 for which the ﬁrst return time τ B ( x,r ) ( x ) of x to the neighborhood B ( x,r ) behaves like 1 /r t , has full Hausdorﬀ dimension on any open set, i.e. dim for any open set G with G ∩ K (cid:7) = ∅ . As a consequence we deduce that the so-called multifractal formalism fails comprehensively for the ﬁrst return time multifractal spectrum. Another application of our results concerns the construction of a certain class of Darboux functions.


Introduction and Statement of Results.
Let X be a set and let S : X → X be a map from X into itself. For U ⊆ X and x ∈ X, let τ U (x) denote the first return time of x to U , i.e.
Poincaré's classical recurrence theorem tells us that if S : X → X a measure preserving map on a probability measure space (X, µ), then µ-a.a. points are infinitely recurrent to any set U ⊆ X with positive measure. In particular, this implies that if X is a metric space and the ball B(x, r) with center x and radius r > 0 has positive measure, then τ B(x,r) (x) < ∞ for µ-a.a. x ∈ X. In view of this, it is natural to ask for quantitative results regarding the dependence on r of the first return time τ B(x,r) (x) of x to the ball B(x, r), i.e. we ask the following question: With which rate does a point x return to a small neighborhood B(x, r) of itself? (1.1) This question has recently attracted an increasing interest and certain results have been obtained within the past 10 years. In the early 1990's Ornstein & Weiss [13] proved that if X is the full shift equipped with the natural metric and S : X → X is the shift map, then for any ergodic shift invariant probability measure µ on X we have for µ-a.a. x ∈ X, where dim loc (µ, x) denotes the local dimension of µ at x, i.e. dim loc (µ, x) = lim r 0 log µB(x, r) log r , (1.3) and where h(µ) denotes the entropy of µ; this result has later been generalized to more general dynamical systems [16]. The reader is referred to [1,4,8] for a different approach to the study of question (1.1). The result in (1.2) immediately gives a lower bound for the Hausdorff dimension of the set of points x for which the first return time τ B(x,r) (x) of x to the neighborhood B(x, r) behaves like 1/r h(µ) for small values of r, i.e. the set Indeed, let dim denote the Hausdorff dimension, and for a probability measure µ on X, let dim µ denote the Hausdorff dimension of µ, i.e. dim µ = inf µ(E)=1 dim E.
The first inequality in (1.4) follows immediately from (1.2), and the second equality in (1.4) follows from the so-called mass distribution principle (cf. [3, p. 55, Theorem 4.2]) and the fact that dim loc (µ, x) = h(µ) for µ-a.a. x by the Shannon-MacMillan-Breiman theorem. Unfortunately, inequality (1.4) only provides a partial and incomplete answer to problem (1.1). The purpose of this paper is to give a comprehensive answer to question (1.1). We now present a few selected examples of the results in this paper. These examples illustrate the surprisingly rich and intricate structure first return times exhibit. Let N be a positive integer and let S : [0, 1] → [0, 1] be defined by S(x) = Nx mod 1. We prove that for any t > 0, the set of points x ∈ [0, 1] for which the first return time τ B(x,r) (x) of x to the neighborhood B(x, r) behaves like 1/r t for small values of r, i.e. those x ∈ [0, 1] such that In fact, we prove a significantly more general and surprising result. For a function ϕ : (0, ∞) → R, let A(ϕ(r)) denote the set of accumulation points of ϕ(r) as r 0, i.e.
A(ϕ(r)) = x ∈ R there exists a sequence (r n)n with r n 0 such that ϕ(r n ) → x . (1.5) Even more surprisingly, we show that for any compact subinterval I of (0, ∞), the set of points x ∈ [0, 1] for which the set of accumulation points of The result in Theorem 1 clearly has a multifractal flavor. In the study of geometric properties of dynamical systems or "fractal" measures one is often interested in the asymptotic behavior of various local quantities associated with the underlying dynamical or geometric structure. For example, one is often interested in the local dimension lim r 0 log µ(B(x,r)) log r of a measure µ (cf. (1.3)) or the ergodic average of a continuous function. These quantities provide a description of various aspects of measures or dynamical systems, e.g. chaoticity, sensitive dependence, et.c. All these quantities provide important information about the underlying geometric or dynamical structure. This idea leads to the notion of multifractal spectra. For a metric space X and and a set Y , we define the multifractal spectrum of a map ϕ : where dim denotes the Hausdorff dimension, cf. [9,10,11]. Motivated by this we define the first return time spectrum of a self map S : X → X on a metric space X by Typically the multifractal spectrum f of a map ϕ : X → Y satisfies the so-called multifractal formalism, i.e. f is strictly concave and there exists a "natural" function τ : Y → R such that f equals the Legendre transform of τ , cf. [9,10,11,17] and the references therein. Theorem 1 shows that the first return time spectrum, in general, fails the multifractal formalism comprehensively: the first return time spectrum is not strictly convex, and since the first return time spectrum is constant, it is easily seen that there is no function τ : R → R such that the first return time spectrum equals the Legendre transform of τ . Another application of Theorem 1 concerns the construction of a very irregular Darboux function of Baire class 3. Recall the a function ϕ : [0, 1] → R is called Darboux if it has the intermediate value property. Also recall that a function ϕ : [0, 1] → R is said to be of Baire class 0 if it is continuous, and that ϕ is said to be of Baire class n for some positive integer n if it is the pointwise limit of a sequence of functions of Baire class n − 1. Intuitively, one should think of a function of Baire class n as being n steps away from being continuous. Now, let N be a positive integer and define S :  Unfortunately, we have not been able to show that the function constructed in the proof of Corollary 3 is of Baire class 2, and we do not know if there exists a function of Baire class 2 satisfying the condition in Corollary 3. (Of course, no function of Baire class 0 (i.e. a continuous function) can satisfy the condition in Corollary 3. Also, since any function of Baire class 1 has a dense set of continuity points, no function of Baire class 1 can satisfy the condition in Corollary 3.) We therefore ask the following question. We emphasize that the above results are particular cases of the theory developed in this paper. In fact, our main results will be formulated in the setting of selfconformal sets. In Section 1.1 we define self-conformal sets and in Section 1.2 we state our main results. The proofs are given in Sections 2-6.
If ω ∈ Σ n , we define the cylinder [ω] generated by ω by

First return time spectra.
In this section we state our main results. We will frequently assume that the list V, X, (S i ) i=1,... ,N satisfies certain "disjointness" conditions, viz. the Open Set Condition (OSC) or the Strong Separation Condition (SSC) defined below.

The Open Set Condition (OSC):
There exists an open non-empty and bounded

The Strong Separation Condition (SSC): There exists an open non-empty and bounded subset
Clearly the SSC implies the OSC. If the SSC is satisfied then it is easily seen that there exists a continuous map S : K → K such that the diagram below commutes . In this case we consider the first return time defined with respect to the map S. However, if the SSC is not satisfied there is not necessarily a continuous map S : K → K that makes the diagram in (1.9) commutative. In this case we consider the natural analogue of τ B(x,r) (x) in the shift space Σ N . Namely, we consider the first return time of ω ∈ Σ N with respect to the shift map. The return times τ [ω|n] (ω) are significantly easier to analyze than their "geometrically" defined counterparts τ B(x,r) (x). Also, it is easily seen that if the SSC is satisfied, then the return times τ B(x,r) (x) and τ [ω|n] (ω) are comparable for all x = π(ω) (cf. Section 6 for details), and results for the "symbolically" defined return times τ [ω|n] (ω) can therefore be used to obtain precise information about the "geometrically" defined counterparts τ B(x,r) (x). In fact, in the main results below we will employ this fact. Indeed, we first state results for the "symbolically" defined return times τ [ω|n] (ω) assuming the OSC and then, using these results, we obtain information about their "geometrically" defined counterparts τ B(x,r) (x) assuming the SSC (2) Assume that the SSC is satisfied. For all t > 0 we have In fact, we obtain significantly more general results providing precise information about the rate at which the first return exponent − log r diverges as r 0. Recall the notation from (1.5), i.e. if ϕ : (0, ∞) → R is a real valued function, then A(ϕ(r)) denotes the set of accumulation points of ϕ(r) as r 0. Below we will use a similar notation for the set of accumulation points of a sequence, i.e. if (x n ) n is a sequence of real numbers then A((x n ) n ) denotes the set of accumulation points of (x n ) n . Using this notation it is seen that and it is therefore natural to study the sets for closed subsets F of R.

Theorem 6.
(1) Assume that the OSC is satisfied. For all compact subintervals I of (0, ∞) and for all open sets G with (2) Assume that the SSC is satisfied. For all compact subintervals I of (0, ∞) and for all open sets G with We do not know if the results in Theorem 6 remain true if the compact subinterval I of (0, ∞) is replaced by an arbitrary closed subset F of (0, ∞), and we therefore ask the following question. . ω n ∈ Σ n , then we write S ω = S ω1 • · · · • S ωn and K ω = S ω K. Proposition 2.1 and Proposition 2.2 state that a conformal iterated function system distorts the geometry of sets in R d in a uniformly bounded way. Both results are standard and their proofs can be found in many papers, cf. for example [2,7,14]. For a conformal iterated function system V, X, (S i ) i=1,... ,N , we define the map Φ : Σ N → R as follows .. ,N be a conformal iterated function system. Then there exists a constant c ≥ 1 such that the following statements hold.

Proof of Theorem 6.(1): construction of the set Z. Write
We must now prove that dim K ≤ dim(G ∩ π(E)). Fix δ > 0. The idea behind the proof is to construct a set Z ⊆ Σ such that and This proves Theorem 6.(1) since δ > 0 is arbitrary. In this section we construct the set Z. In Section 4 we prove that Z ⊆ E and π(Z) ⊆ G (Proposition 4.3), and finally in Section 5 we prove that dim K − 2δ ≤ dim π(Z) (Proposition 5.2).
Let c denote the constant in Proposition 2.1. Observe that it follows from the principle of bounded variation (Proposition 2.1) that if n is a positive integer, then Similarly we obtain Letting n → ∞ yields This implies that P (s M Φ) → 0. Since the function t → P (tΦ) is strictly decreasing and continuous with P (sΦ) = 0, we therefore conclude that s M → s.

By Proposition 3.1 we can choose a positive integer M such that
Next, letν denote the Gibbs state of the Hölder continuous function s M Φ M , i.e. ν is the unique Borel probability measure on Γ N M for which there exists a constant C > 0 such that for all positive integers n and all ω ∈ Γ N M . To complete the construction of the set Z we need the following elementary lemma.

Lemma 3.2.
Let 0 < a ≤ b < ∞ and let (t n ) n be a sequence of real numbers such that a ≤ t n ≤ b and |t n+1 − t n | ≤ a n+1 for all n. Then there exists a sequence (k n ) n of integers such that: (1) kn n p → ∞ for all p > 0;
It follows from Proposition 2.2 that it suffices to show that (4.1) We will now prove (4.1). Fix a positive integer n, and choose (unique) positive integers m = m(n) and k = k(n) with 0 ≤ k ≤ j m+1 such that Let c be the constant in Proposition 2.1. The choice of m and k, and the principle of bounded variation (Proposition 2.1) implies that where J m = j 1 + · · · + j m . However, since ω ∈ Z = Λ(X), we conclude that σ ∈ X. This implies that We also have (using (3.6))
We must now prove that τ [ω|n] (ω) ≥ q n , i.e. we must prove that ω|n = (S k ω)|n for all k < q n . Assume in order to obtain a contradiction that ω|n = (S k ω)|n for some k < q n . We now introduce some terminology. Write for some n i ≥ M , and where τ i does not contain v as a substring and the first and the last letter in τ i are both different from v. In particular, we have i.e. S k+|τ0| ω starts with the string v. For a positive integer m, we will say that S k+|τ0| ω starts inside σ m if and we will say that S k+|τ0| ω starts inside ω m if q m ≤ k + |τ 0 | < q m + m + 1 .
Since n ≥ |γ| + M = |γv|, we conclude that ω|n begins with the string v. Hence (S k ω)|n = ω|n also begins with the string v. Moreover, since each σ i begins and ends with a letter different from v (namely u) and since the string v does not appear as a substring in any of the strings σ 1 , σ 2 , . . . , σ n , we conclude that S k ω starts inside ω m for some m < n. Write  Since S k ω starts inside ω m and (S k ω)|m = ω|m begins with the substring τ 0 v we conclude that for some j > 1, (4.5) cf. Figure 1.

Proof.
We first prove that Z ⊆ E. For ω ∈ Z, Proposition 4.1 and Proposition 4.2 imply that 1 n log diam K ω|n → −χ and 1 n log τ [ω|n] (ω) − χt n → 0. It follows easily from this and the fact that 0 < inf n t n ≤ sup n t n < ∞, that Finally, since {t n | n ∈ N} is a dense subset of I, this implies that This shows that Z ⊆ E. Next, we prove that π(Z) ⊆ G. Indeed, since Z ⊆ [γ], it follows from the choice of γ (cf. (3.10)) that π(Z) ⊆ π([γ]) ⊆ K γ ⊆ G.

5.
Proof of Theorem 6.(1): dim K − 2δ ≤ dim π(Z). We will now prove that dim K − 2δ ≤ dim π(Z) . (5.1) We will prove (5.1) by constructing a probability measure µ on π(Z) for which there exists a constant C > 0 such that for all x ∈ π(Z) and all r > 0. Recall thatν is the Gibbs state of s M Φ M (cf. (3.5)), and recall that the map Λ : Γ N M → Σ N is defined in (3.11). Now define the probability measureμ on Σ N bỹ and finally define the probability measure on K by The next lemma is a standard result due to Hutchinson [5].
(2) There exists a constant C > 0 such that for all x ∈ π(Z) and all r > 0.