RETURN TIME STATISTICS VIA INDUCING

H. BRUIN, B. SAUSSOL, S. TROUBETZKOY, AND S. VAIENTI

Abstract. We prove that return time statistics of a dynamical system do not change if one passes to an induced (i.e. first return) map. We apply this to show exponential return time statistics in i) smooth interval maps with nowhere–dense critical orbits and ii) certain interval maps with neutral fixed points. The method also applies to iii) certain quadratic maps of the complex plane.

1. Introduction

In the last few years the study of return and hitting times has become an important ingredient for the statistical characterization of dynamical systems. A historical account of this approach can be found in the review paper [10] or in the introduction of [20], where an extended bibliography is provided. To pose the problem in general terms, suppose that T is a measure preserving transformation of a measure space (X, μ) and U_{x} ⊂ X is a neighborhood of a point x ∈ X. The two questions which are the fundamental objects of investigation are:

1. What is the probability distribution of the first hitting time of the set U_{x} as μ(U_{x}) → 0?

2. What is the probability distribution of the first return time for points leaving from U_{x} as μ(U_{x}) → 0?

For large classes of dynamical systems showing some sort of hyperbolic behavior, the answers to those questions are surprisingly easy and universal. The probability distribution function up to a suitable normalization, turns out to be the exponential 1-law exp(−t): If τ_{U}(x) is the smallest integer n such that T^{n}(x) ∈ U (so τ stands for first hitting and for first return time), then both

μ(\{x ∈ X; τ_{U}(x) ≥ \frac{t}{μ(U)}\}) → e^{−t}

2000 Mathematics Subject Classification. Primary: 37B20 - Secondary: 37D35, 37F10.

Key words and phrases. return time, exponential statistics.

HB was supported by the Royal Dutch Academy of Sciences (KNAW) and the PRO-DYN program of the European Science Foundation. Also the hospitality of the Centre Physique Théorique, Luminy, is gratefully acknowledged.

BS was supported by FCT’s Funding Program and by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, Lisbon, Portugal.

and

\frac{1}{μ(U)}μ(\{x ∈ U; τ_{U}(x) ≥ \frac{t}{μ(U)}\}) → e^{−t},

as μ(U) → 0. After the works of Pitskel [36], Hirata [19] and Collet [11] at the beginning of the nineties which focused essentially on uniformly hyperbolic dynamical systems, new developments have brought at least three improvements:

• The possibility to treat larger classes of systems, notably certain non– uniformly hyperbolic systems;

• The possibility to estimate the error to the asymptotic distribution which is closely related to the hyperbolic character of the transformation and to the influence of sets with short recurrence;

• The application of the exponential statistics to the computation of the fluctuations of repetition times in the Ornstein-Weiss formula for the metric entropy.

These improvements were the result of four new approaches. The first approach was originated by Galves and Schmitt [15] and was formulated in a probabilistic setting for systems satisfying a φ-mixing condition. This approach was translated into the dynamical systems language by Haydn for Julia sets [16], by Paccaut [34] for a large class of non-Markovian maps of the interval and by Boubakri [4] for some Collet-Eckmann unimodal maps. Recently Abadi [2] extended it for α-mixing stationary processes. The true probabilistic flavor of this technique led to the discovery of a close connection with the Ornstein-Weiss [33] theorem on metric entropy and resulted in a proof of the log-normal fluctuations of the repetition times for Gibbs measures in [13] (see [22] for related results); see also [34] for further developments.

The second approach is due to Hirata, Saussol and Vaienti [20]. First for measure-preserving dynamical systems it quantifies the error to the asymptotic exp(−t) distribution giving precise bounds in terms of the mixing properties of the systems and of a sharp control of short recurrences (this last point is in itself an interesting subject suggesting the possibility to formulate a thermodynamics of return times [39, 40]). These bounds are then computed for a large class on non-uniformly hyperbolic maps of the interval in [20]. We used this method in the present paper to prove the exponential statistics for the class of Rychlik maps introduced in Section 3. A key observation in [20] allows to link the statistics of hitting and return times quoted above, namely the distribution of the first return time is close to the exponential law if and only if it is close to the distribution of the first hitting time.

The third approach goes back to a probabilistic paper of Sevast’yanov [41]. This approach was already used by Pitskel [36] in the context of Markov chains and Axiom-A diffeomorphisms. The same scheme was applied by Haydn in [16] and [17] for equilibrium states on Julia sets, in the presence of a supremum gap in the Perron-Frobenius operator (namely, if f is an Hölder continuous function on the Julia set of a rational map of degree at least 2, then P (f) > sup f, where P (f) is the topological pressure of f). The original technique of Sevast’yanov only allowed to prove the convergence to the exponential 1-law. A recent work by Haydn and Vaienti [18] quantifies this technique and thus provides bounds for the error estimate in the case of rational maps and parabolic maps of the interval; such bounds improve all other existing bounds for the class of maps considered.

The last approach has been introduced by Collet [12]; it is an application of the Gumbel’s law for entrance times [14] to a large class of non–uniformly hyperbolic dynamical systems with exponential decay of correlations for which a tower satisfying Young’s conditions can be constructed [48, 49]. Collet has proven that the statistics of closest return to a given point is almost surely asymptotically Poissonian and gave the fluctuations for the nearest return to the starting point.

It is useful to remark at this point that the approaches we are describing also permit to compute the probability distribution of successive visits to the set U_{x}: it gives, in all the cases where the exponential statistics holds, the Poisson distribution \frac{t^{n}}{n!} e^{−t}, where n is the number of visits of U_{x}.

In this paper we propose a new scheme which allows us to compute the asymptotic distributions for the first and successive return times whenever these distributions are known on a subset endowed with an induced structure. Its interest lies in the fact that several non-uniformly hyperbolic systems admit around almost every point (w.r.t. the invariant measure) a neighborhood where the first return map acts as the induced transformation and exhibits hyperbolic behavior. The first application of our approach is for C^{2} interval maps; we prove the exponential return time statistics under the additional hypothesis that the map preserves a conformal measure (see Section 4 for details) and that the closure of the orbit of the critical points has zero measure. Our result complements those given Boubakri [4] and by Collet [12], in the sense that, contrarily to them, the growth rate of the derivatives along the critical orbit plays no role in our theorem. It applies for example to the well known Fibonacci map [28].

As a second example, we improve the exponential statistics for the parabolic map studied in [20]. In both cases, the first return map of the induced systems belongs to a class of piecewise monotonic maps of the interval (with countably many pieces), previously investigated by Rychlik; the induced measure shows exponential decay of correlations with respect to the Rychlik map and this is the kind of hyperbolic behavior which is sufficient to prove the exponential statistics for such maps.

In most of the approaches quoted above, the set U_{x} shrinking to {x} was chosen in the class of cylinders generated by some partition of the space. The exponential return time statistics around cylinders is what is needed to compute the fluctuations in the Ornstein-Weiss theorem [39].

Instead of cylinders one can also use balls for the sets U_{x}. The use of balls has a twofold interest. Firstly, it enlarges the class of sets where to check the recurrence of rare events, such as entrance time (particularly from the perspective of numerical simulations in physical situations, see for example [9] and [50]), Second, the behavior of return time in balls is strongly related to other statistical indicators, like local dimension and Lyapunov exponents, as shown in [40].

Balls instead of cylinders were used by Haydn [17] and Collet [12]. In this article we prove exponential statistics for Rychlik maps around balls. This improves the work of [34], who considered a particular class of Rychlik maps (the covering weighted systems introduced in [25]), and only cylinders around points.

The general scheme proposed in this paper could in principle be applied to a wide class of dynamical systems for which the induced transformation enjoys an exponential statistics; we think in particular to rational maps of the Riemann sphere (an example is given in Chapter 5) and of billiards for which the induced structures, although not yet studied, seem however more accessible than the original systems.

As a final observation, we would like to point out the robustness of the exponential statistics, which persists in the non-hyperbolic systems covered in this paper. This confirms its central role in the characterization of recurrence for ergodic systems and motivate further analysis for larger class of transformations especially in dimension greater than one and exhibiting singularities.

2. Statistics via inducing: an abstract theorem

Suppose that (T, X, μ) is an ergodic measure preserving transformation of a smooth Riemannian manifold X. Let X̂ ⊂ X be an open set. Furthermore let T̂ : X̂ → X̂ be the first return map. We denote the induced measure by μ̂: the measure preserving transformation (T̂, X̂, μ̂) is therefore ergodic.^{1} For z ∈ X we denote by U_{r}(z) the ball of radius r centered at z and by τ_{U_{r}(z)} (resp. τ̂_{U_{r}(z)}) the first return time of U_{r}(z) for T (resp. T̂). For a μ (resp. μ̂) measurable set A we denote by μ_{A} (resp. μ̂_{A}) the induced measure on the set A. We suppose that (T̂, X̂, μ̂) has return time statistics f̂(t): i.e. for μ̂-a.e. z ∈ X̂, ∃ε_{z}(r) > 0 with ε_{z}(r) → 0 as r → 0 such that

sup_{t≥0} |μ̂_{U_{r}(z)}(x ∈ U_{r}(z) : τ̂_{U_{r}(z)}(x) > \frac{t}{μ̂(U_{r}(z))}) − f̂(t)| < ε_{z}(r). (1)

Our main theorem of this section is that on X̂ the map T̂ obeys the “same” statistical law as T:

Theorem 2.1. If f̂ is continuous at 0, then there exists f : ℝ^{+} → [0, 1] such that N =^{def} {x : f ≠ f̂} is countable and for μ–a.e. z ∈ X̂ and t ∉ N, there exists δ_{z,t}(r) > 0 with δ_{z,t}(r) → 0 uniformly in t as r → 0 such that

1We could weaken our assumptions by demanding that the map T̂ is ergodic instead of T; the ergodicity of T is however recovered when X = ∪_{n=0}^{∞}T^{−n}X̂

|μ_{U_{r}(z)}(x ∈ U_{r}(z) : τ_{U_{r}(z)}(x) > \frac{t}{μ(U_{r}(z))}) − f(t)| < δ_{z,t}(r).  (2)

In addition, if f̂ is continuous then the convergence is uniform in t, which means there exists δ_{z}(r) → 0 as r → 0 such that for all t ∈ ℝ^{+} and r > 0, δ_{z,t}(r) ≤ δ_{z}(r).

\underline{Remark:} the function f̂ (resp. f) is known to be decreasing and thus continuous everywhere except for an at most countable set of exceptional points. The set N is a subset of the points of discontinuity of f̂.

Proof. First of all, if μ has an atom then μ is supported on a periodic orbit, hence the result is trivial because in this case f(t) = f̂(t) = 1 for t ≤ 1 and 0 otherwise. We may assume then that μ has no atoms, consequently μ(U_{r}(z)) → 0 as r → 0 for all z ∈ X̂.

At first we suppose that z ∈ X̂ and assume that r is small enough that U_{r}(z) ⊂ X̂. Note that this implies μ̂_{U_{r}(z)} = μ_{U_{r}(z)}.

For x ∈ X̂ let n(x) =^{def} τX̂(x) be the T –first return time of x to X̂. By Kac’s Theorem and ergodicity of μ̂ we have

A_{m}(x) =^{def} \frac{1}{m} ∑_{i=0}^{m−1} n(T̂^{i}x) ⟶_{m→∞} c =^{def} ∫X̂ n(x) dμ̂ = \frac{1}{μ(X̂)}  (3)

for μ̂-a.e. x ∈ X̂. Let G =^{def} {x ∈ X̂ : lim_{m→∞} A_{m}(x) = c}. Clearly μ̂(G) = 1. For all x ∈ G and for all ε > 0 there exists m(x, ε) < ∞ such that

|A_{m}(x) − c| < ε for all m ≥ m(x, ε). Let G_{m} =^{def} {x ∈ G : m(x, ε) < m}

where we have suppressed the obvious ε dependence on the set G_{m}. We choose M =^{def} M(ε) such that μ(G_{M}) > 1 − ε.

By definition | ∑_{i=0}^{m−1} n(T̂^{i}x) − cm| < εm for all m ≥ M and all x ∈ G_{M}. Thus T̂^{m}(x) = T^{cm+s}(x) for some s = s(x) with |s| < εm. It immediately follows that τ_{U_{r}(z)}(x) = cτ̂_{U_{r}(z)}(x) + s with |s| < ετ̂_{U_{r}(z)}(x) whenever τ̂_{U_{r}(z)}(x) ≥ M and x ∈ G_{M}.

Next we define

G̃_{M} =^{def} {z ∈ G_{M} : μ_{U_{r}(z)}(G_{M}) ⟶_{r→0} 1}.  (4)

By the Lebesgue Density Theorem, μ̂(G̃_{M}) = μ̂(G_{M}) and thus μ̂(G̃_{M}) > 1− ε. For each z ∈ G_{M} there exists r(z, M, ε) > 0 such that μ_{U_{r}(z)}(G_{M}) > 1 − ε for all r < r(z, M, ε). Thus if R > 0 is sufficiently small then μ̂(G_{M,R}) > 1−ε,

where G_{M,r} =^{def} {z ∈ G̃_{m} : r(z, M, ε) > r}.

Let S be the set of points z ∈ X̂ for which ε_{z}(r) → 0 as r → 0. By assumption μ̂(S) = 1. Hence μ̂(G_{M,R} ∩ S) = μ̂(G_{M,R}) > 1 − ε.

Denote

F_{U_{r}(z)}(t) =^{def} μ_{U_{r}(z)}({x ∈ U_{r}(z) : τ_{U_{r}(z)} > t/μ(U_{r}(z))}),
F̂_{U_{r}(z)}(t) =^{def} μ̂_{U_{r}(z)}({x ∈ U_{r}(z) : τ̂_{U_{r}(z)} > t/μ̂(U_{r}(z))}),

and set B_{U_{r}(z)}(M) = {x ∈ U_{r}(z) : τ̂_{U_{r}(z)}(x) > M}.

Assume additionally that z ∈ G_{M,R} ∩ S. The limiting distribution f̂(t) is continuous at 0 and f̂(0) = 1, hence if r is sufficiently small

μ_{U_{r}(z)}(B_{U_{r}(z)}(M)^{c}) = μ̂_{U_{r}(z)}(B_{U_{r}(z)}(M)^{c}) ≤ 1 − f̂(Mμ̂(U_{r}(z))) + ε_{z}(r) < ε.

Thus for all t ∈ [0, ∞),

F_{U_{r}(z)}(t) ≤ μ_{U_{r}(z)} (G_{M,R}^{c} ∪ B_{U_{r}(z)}(M)^{c}) +

+μ_{U_{r}(z)} (G_{M,R} ∩ B_{U_{r}(z)}(M) ∩ \{x : τ_{U_{r}(z)}(x) > \frac{t}{μ(U_{r}(z))}\})

≤ μ_{U_{r}(z)}(G_{M,R}^{c}) + μ_{U_{r}(z)}(B_{U_{r}(z)}(M)^{c}) +

+μ_{U_{r}(z)} (G_{M,R} ∩ B_{U_{r}(z)}(M) ∩ \{x : τ̂_{U_{r}(z)}(x) > \frac{t/(c + ε)}{μ(U_{r}(z))}\})

≤ 2ε + F̂_{U_{r}(z)} (\frac{t}{1 + ε/c}) .

A similar computation with 1 − F_{U_{r}(z)}(t) = μ_{U_{r}(z)}(τ_{U_{r}(z)} ≤ t/μ(U_{r}(z))) yields

F̂_{U_{r}(z)} (\frac{t}{1 − ε/c}) − 2ε ≤ F_{U_{r}(z)}(t) ≤ F̂_{U_{r}(z)} (\frac{t}{1 + ε/c}) + 2ε. (5)

For r sufficiently small, ε_{z}(r) < ε. Thus inequality (5) implies:

−3ε + f̂(\frac{t}{1 + ε/c}) ≤ F_{U_{r}(z)}(t) ≤ f̂(\frac{t}{1 + ε/c}) + 3ε.  (6)

Define

δ_{t}(z) =^{def} max \{|f̂(t) − f̂(\frac{t}{1 − c^{−1}ε})| , |f̂(t) − f̂(\frac{t}{1 + c^{−1}ε})|\} . (7)

For each point t of continuity of f̂, the function δ_{t}(ε) → 0 as ε → 0. Combining (6) and (7) yields:

|F_{U_{r}(z)}(t) − f̂(t)| ≤ 3ε + δ_{t}(ε).  (8)

We just showed that for any ε > 0, there exists a set G(ε) with μ̂(G(ε)) > 1 − ε and a real number R(ε) > 0 such that for all z ∈ G(ε), Inequality (8) holds whenever r < R(ε). The conclusion of the theorem for μ-a.e. z ∈ X̂ follows from the remark that μ-almost every z ∈ X̂ is contained in the set of full measure ∩_{n>0} ∪_{m>n} G(1/m).

Finally we want to prove the uniform convergence in t in the case f̂ is continuous. Since f̂(t) → 0 as t → ∞, it is uniformly continuous; hence q(δ) = sup_{0≤s<t<s+δ} |f̂(s) − f̂(t)| → 0 as δ → 0.

Moreover, f̂(t) is bounded by 1/t by Chebychev’s inequality. Hence Equation (7) gives

δ_{t}(ε) ≤ min (q (\frac{tε}{c − ε}) , 1/t) + 3ε.

If tε/(c − ε) < \sqrt{ε} then δ_{t}(z) ≤ q(\sqrt{ε}) + 3ε, while if tε/(c − ε) ≥ \sqrt{ε} then δ_{t}(z) ≤ 1/t ≤ \sqrt{ε}/(c − ε) + 3ε.

\underline{Remark:} Using the same method it is possible to show the counterpart of Theorem 2.1 for the successive return times and the number of visits.

3. Piecewise monotonic transformations

In this section we show that piecewise monotonic maps of Rychlik’s type (even with countably many monotonic pieces) enjoy exponential statistics. Let X ⊂ ℝ be a compact set and m a Borel regular probability measure on X. The variation of a function g : X → ℝ is defined by

Var g =^{def} sup{∑_{j=0}^{k−1} |g(x_{j+1}) − g(x_{j})|},

where the supremum is taken along all finite ordered sequences (x_{j})_{j=1,... ,k} with x_{j} ∈ X. The norm ∥g∥_{BV} = sup |g| + Var g makes BV = {g : X → ℝ : ∥g∥_{BV} < ∞} into a Banach space. We endow X with the induced topology and denote by B(X) the Borel σ-algebra of X. We say that I ⊂ X is an X-interval if there exists some interval J ⊂ ℝ such that I = X ∩ J.

Definition 3.1 (R-map). Let T : Y → X be a continuous map, Y ⊂ X open and dense, and m(Y) = 1. Let S = X \ Y. We call T an R-map if the following is true:

1. There exists a countable family Z of closed X-intervals with disjoint interiors (in the topology of X) such that ∪_{Z∈Z}Z ⊃ Y and for any Z ∈ Z the set Z ∩ S consists exactly of the endpoints of Z;

2. For any Z ∈ Z, T |_{Z∩Y} admits an extension to a homeomorphism from Z to some X-interval;

3. There exists a function g : X → [0, ∞), with Var g < +∞, g|_{S} = 0 such that the operator P : L^{1}(m) → L^{1}(m) defined by

P f(x) = ∑_{y∈T^{−1}(x)} g(y)f(y)

preserves m. In other words, m(P f) = m(f) for each f ∈ L^{1}(m), that is m is g^{−1}-conformal;

4. T is expanding: sup_{x∈X} g(x) < 1.

We first remark that if T is an R-map, then any iterate T^{n} is also an R-map (see Lemma 2 in [38] and the discussion before for details). Given a weight function g such that sup_{X} g < 1 and Var g < ∞, Liverani, Saussol and Vaienti have shown in [25] the existence of a conformal measure m which fulfills the hypotheses, under the additional assumption that T is covering:

for any interval I ⊂ X, there exists an integer N > 0 such that

inf_{x∈Y} P^{N}χ_{I}(x) > 0.

Proposition 3.1. Suppose that X = [0, 1], m is Lebesgue measure and T is a piecewise monotonic transformation on X. If

1. T |_{Z∩X} is C^{2} for each Z ∈ Z,

2. T is uniformly hyperbolic, i.e. for some M > 0, inf |(T^{M})′| > 1, the infimum being taken on the subset of X where (T^{M})′ is defined,

3. T has bounded distortion, which means

sup_{Z∈Z} sup_{Z} \frac{|T′′|}{|T′|^{2}} < ∞ and ∑_{Z∈Z} sup_{Z} \frac{1}{|T′|} < ∞,  (9)

then T is an R-map with weight function g = 1/|T′| on Y and g = 0 on S.

Proof. It is enough to check that the variation of g = 1/|T′| is finite. Since g is C^{1} on the interior of Z ∈ Z, the variation is estimated by

Var g ≤ ∑_{Z∈Z} ∫_{Z} |g′(t)|dt + 2 ∑_{Z∈Z} sup_{Z} g

≤ ∑_{Z∈Z} ∫_{Z} \frac{|T′′(t)|}{|T′(t)|^{2}} dt + 2 ∑_{Z∈Z} sup_{Z} \frac{1}{|T′|}

≤ sup_{Z∈Z} sup_{Z} \frac{|T′′|}{|T′|^{2}} + 2 ∑_{Z∈Z} sup_{Z} \frac{1}{|T′|} < ∞.

R-maps possess very strong statistical properties, which we collect below

Theorem 3.1 (Rychlik [38]). If T is an R-map then there exists an invariant measure μ which is absolutely continuous with respect to the conformal measure m with density h = \frac{dμ}{dm} ∈ BV. The measures m and μ have no atoms. In addition, there exists a partition (mod m) of X into disjoint open sets X_{i,j}, where i = 1, …, k and j = 1, …, L_{i} such that each system (T^{L_{i}}|_{X_{i,j}}, X_{i,j}, μ_{X_{i,j}}) is mixing and has exponential decay of correlations for bounded variation observable, which means that for some C ∈ (0, ∞) and θ ∈ (0, 1)

|∫φ ∘ T^{nL_{i}}ψdμ_{X_{i,j}} − ∫φdμ_{X_{i,j}} ∫ψdμ_{X_{i,j}} | ≤ C∥φ∥_{L^{1}(m)}(∥ψ∥_{BV})θ^{n} (10)

for any φ ∈ L^{1}(m) and ψ ∈ BV.

This theorem tells us in particular that X can be partitioned (mod m) into compact sets X_{i,j} where some iterate of T is again an R-map with a mixing measure equivalent to m|_{X_{i,j}} and which has exponential decay of correlations for bounded variation observable.

Theorem 3.2. Any R-map T with conformal measure m and invariant mixing measure μ ≪ m has exponential return time statistics around balls.

Proof. In the proof we use the estimate given by Lemma 2.4 in [20] to apply Theorem 2.1 in [20]. We recall the quantities considered there (N is any integer).

a_{N}(U) =^{def} μ_{U}({x : τ_{U}(x) ≤ N}),
b_{N}(U) =^{def} sup {|μ_{U}(T^{−N}V) − μ(V)|: V is measurable} ,
c(U) =^{def} sup_{k≥0} |μ_{U}({x : τ_{U}(x) > k}) − μ({x : τ_{U}(x) > k})|.

We denote by h the density of the measure μ with respect to m. Let U ⊂ X be an interval in X and set τ(U) =^{def} inf{τ_{U}(x) : x ∈ U}. We now compute an upper bound for a_{N} and b_{N}.

Obviously a_{N}(U) = ∑_{n=τ(U)}^{N} μ_{U}({x : τ_{U}(x) = n}), and for each n ≥ τ(U)

μ_{U}(\{x : τ_{U}(x) = n\}) ≤ \frac{1}{μ(U)} ∫ 1I_{T^{−n}U}1I_{U}dμ
= ∫ (\frac{1I_{U}}{μ(U)}) ∘ T^{n} · (1I_{U} − μ(U))dμ + μ(U)
≤ C \frac{m(U)}{μ(U)} ∥1I_{U} − μ(U)∥_{BV} θ^{n} + μ(U).

We used for the last inequality the estimate on decay of correlations (10) given by Theorem 3.1, with φ = 1I_{U}/μ(U) and ψ = 1I_{U} − μ(U). Since U is an interval we have ∥1I_{U} − μ(U)∥_{BV} ≤ 3, hence the summation on n = τ(U), …, N gives

a_{N}(U) ≤ \frac{3C}{1 − θ} \frac{m(U)}{μ(U)}θ^{τ(U)} + Nμ(U).  (11)

We consider now b_{N}(U). The decay of correlations given by (10) yields (with φ = 1I_{V} and ψ = 1I_{U}/μ(U))

b_{N}(U) ≤ sup_{V ∈B(X)} C∥1I_{V} ∥_{L^{1}(m)}∥1I_{U}∥_{BV} θ^{N}/μ(U) ≤ 3C \frac{θ^{N}}{μ(U)}. (12)

Lemma 2.4 in [20] together with (11) and (12) yield (with N = \frac{2 log μ(U)}{log θ})

c(U) ≤ c_{1} \frac{m(U)}{μ(U)}θ^{τ(U)} + c_{2}μ(U) + c_{3}μ(U)| log μ(U)|,  (13)

for some constants c_{1}, c_{2} and c_{3} independent of the interval U.

Since μ has no atoms the countable union W = ∪_{j=0}^{∞}T^{−j}S has zero measure. Hence for μ-a.e. point z ∈ X, the iterates T^{k}z are well defined and T^{k}z ∉ S. One easily sees then that τ(U_{r}(z)) → +∞ as r → 0 provided z is not periodic. Consider the set

G = \{z ∈ X \ W : z is not periodic and D(z) =^{def} \overline{lim}_{r→0} \frac{m(U_{r}(z))}{μ(U_{r}(z))} < +∞\}.

The Lebesgue Density Theorem tells us that D(z) is μ-almost everywhere finite because the density h = \frac{dμ}{dm} > 0, μ-a.e. Since μ is aperiodic (it is ergodic with no atoms), we conclude that μ(G) = 1.

Moreover, for all z ∈ G, using (13) we get c(U_{r}(z)) → 0 as r → 0. We conclude then by Theorem 2.1 in [20] that for μ-almost all z ∈ X,

μ_{U_{r}(z)} (τ_{U_{r}(z)} > \frac{t}{μ(U_{r}(z))}) ⟶_{r→0} e^{−t}

uniformly in t ∈ [0, ∞).

4. Statistic of return times for non-hyperbolic interval maps

Here we prove that a large class of interval maps enjoy exponential statistics for the return time. The strategy is to prove that around almost every point, an interval can be found whose first return map is an R-map.

4.1. Smooth maps with critical points. Let T : [0, 1] → [0, 1] be a C^{2} interval map. Denote the critical set by Crit =^{def} {x ∈ [0, 1] : T′(x) = 0}. For this class of systems, the existence of invariant probability measures and their statistical properties have been frequently studied by means of induced maps (jump transformations rather than first return maps) and tower constructions. The situation is best understood for unimodal map with exponential growth of the derivatives along the orbit of the critical value. This is known as the Collet-Eckmann condition, but a slightly stronger form (involving a slow recurrence condition of the critical point) has been applied by Benedicks & Carleson [3], in which also the prevalence of such behavior in the quadratic family is proven.

Every Collet-Eckmann unimodal map has an absolutely continuous invariant probability measure μ, μ has exponential decay of correlations (w.r.t. to some iterate of the map), satisfies the Central Limit Theorem, and the Perron-Frobenius operator with suitable weights has a spectral gap [47, 21]. An important quantity is the tail behavior of the inducing scheme, i.e. the asymptotic behavior of m({x; R(x) > n}), where m is the reference measure (Lebesgue) and R the inducing time (playing the role of the first return time in our paper). Collet-Eckmann maps have exponential tail behavior: m({x; R(x) > n}) ≤ Ce^{−αn} for some α, C > 0. In [7] the tail behavior of induced maps over multimodal maps with arbitrary growth rates on the critical orbits is computed. The correlation decays were computed using Young’s framework [48].

The multiple return time statistic for Collet-Eckmann maps has been studied (in the Benedicks-Carleson setting) by Boubakri [4]; he finds that they have a Poissonian distribution, and proves additional fluctuation results. A more general approach is taken by Collet [12], but also his result relies essentially on exponential tail behavior of the inducing structure.

The below theorem covers certain maps that are not Collet-Eckmann, and therefore (cf. [32]) have strictly subexponential decay of correlations. One should think of (unimodal) maps with a so-called persistently recurrent critical point, the Fibonacci map being the best known example, see [28]. Such maps frequently admit absolutely continuous invariant probabilities [28, 5], but are never Collet-Eckmann, see [6, Theorem 2]. In this sense, Theorem 4.1 complements Boubakri’s and Collet’s results. The main importance of the theorem is that it shows that, as long as the measure is mixing, exponential, and in fact, any rate of mixing is not needed to have exponential return time statistics.

We will assume that the orbit of the critical set is nowhere dense. This enables us to use a first return map as induced transformation. The only information on the tail behavior we need is that ∑_{n} m({x; τ(x) = n}) < ∞. This is equivalent to the existence of an invariant probability measure μ ≪ m. Furthermore, we state the result for multimodal maps and for |T′|^{t}-conformal measures m_{t}.

Theorem 4.1. Let T : [0, 1] → [0, 1] be as above. Assume that

1. there exists a non-atomic |T′|^{t}-conformal probability measure m_{t} for some t > 0,

2. T preserves an ergodic probability measure μ ≪ m_{t}, and

3. m_{t}(\overline{orb(Crit)}) = 0.

Then (supp(μ), T, μ) has exponential return time statistics.

\underline{Remark:} A special case is t = 1, i.e. m = m_{t} is Lebesgue measure. A point c ∈ Crit is called non-flat if there exists an ℓ, 1 < ℓ < ∞, such that |f(x) − f(c)| = O(|x − c|^{ℓ}). If each critical point is non-flat, then the condition m(\overline{orb(Crit)}) = 0 follows from

orb(Crit) is \{ \begin{matrix}nowhere dense & if #Crit = 1, \\ a minimal set & if 2 ≤ #Crit < ∞. \end{matrix}

This was shown by Martens [29] (see also [30, Theorem V.1.3’]) if #Crit = 1 and by Vargas [45] if 2 ≤ #Crit < ∞.

The case t < 1 comes into view when T has a periodic attractor and the measure m_{t} is supported on a repelling Cantor set.

Proof. Let X = supp(μ) \ \overline{orb(Crit)}. Obviously μ(X) = 1. Let x be any recurrent point in X; by the Poincaré Recurrence Theorem, this concerns μ-a.e. x ∈ X.

Let Y be the component of [0, 1]\\overline{orb(Crit)} containing x and n be any integer such that T^{n}(x) ∈ Y. As orb(Crit) ∩ Y = ∅, there exists a neighborhood U of x such that T^{n} maps U monotonically onto Y. Let T̂ be the first return map to U. By taking n sufficiently large, we can ensure that U is compactly contained in Y and that T^{i}(∂U) ∩ U = ∅ for all i ≥ 0.

Let W ⊂ U be any maximal interval on which T̂|_{W} is monotone. Since orb(∂U) ∩ U = ∅, it follows that T̂ : W → U is onto, and therefore contains a fixed point, say p_{W}. The derivative |T̂′(p_{W})| is uniformly bounded away from 1, see [30, page 268].

Write X̂ = X ∩ U and Z = X ∩ W. Let Z be the partition of X̂ into the sets Z. We will show that T̂ : ∪_{Z∈Z}Z → X̂ is an R-map.

Given intervals I ⊂ J, J is said to contain a δ-scaled neighborhood of I if both components of J \ I have length ≥ δ|I|. Let δ_{0} > 0 be the distance between ∂Y and ∂X. For each branch T̂|_{W} = T^{k}|_{W} (where k =^{def} τ_{U}(W) is the return time), there is an interval W′ ⊃ W such that T^{k} maps W′ monotonically onto Y. It follows that T^{k}(W′) contains a \frac{δ_{0}}{|U|}-scaled neighborhood of T^{k}(W).

More precisely, if x, y ∈ W, then T^{k}(W′) contains a δ =^{def} \frac{δ_{0}}{|T^{k}(x)−T^{k}(y)|}-scaled neighborhood of (T^{k}(x), T^{k}(y)). This allows us to use the Koebe Principle:

Proposition 4.1 (Koebe Principle). Let W′ ⊃ W be intervals and assume that T^{n} : W′ → T (W′) is a C^{2} diffeomorphism. If T^{n}(W) contains a δscaled neighborhood of T^{n}(W), then there exists K such that the distortion

sup_{s,t∈(x,y)⊂W} \frac{T̂′(s)}{T̂′(t)} ≤ K.  (14)

\underline{Remark 1:} Historically, the Koebe Principle was proven under the assumption that T has negative Schwarzian derivative, see [30, Chapter III.6]). Recent work of Kozlovski [23], extended to the multimodal case by Van Strien & Vargas [42] shows that the C^{2} assumption suffices. The magnitude of K = K(δ) = K_{0}(\frac{1+δ}{δ})^{2}, where K_{0} depends only on the map.

\underline{Remark 2:} By taking U small, we can choose K as close to 1 as we want. It follows that inf_{Z∈Z} inf_{Z} |T̂′(x)| ≥ \frac{1}{K} inf_{Z∈Z} |T̂′(p_{W})| > 1, i.e. T̂ is expanding.

Since m_{t} is g^{−1}-conformal,

m_{t}(X̂) = ∫_{Z} |T̂′(x)|^{t}dm_{t} ≤ sup_{Z} |T̂′(x)|^{t}m_{t}(Z).

By (14),

sup_{Z} g = sup_{Z} \frac{1}{|T̂′|^{t}} ≤ K_{0}^{t} inf_{Z} \frac{1}{|T̂′|^{t}} ≤ K_{0}^{t} \frac{m_{t}(Z)}{m_{t}(X̂)}.

This shows that ∑_{Z} sup_{Z} g ≤ K_{0}^{t} < ∞.

Let I ⊃ J be intervals such that T |I is monotone, and let L and R be the components of I \ J. If I is sufficiently small, T expands the cross ratio if

\frac{|T (I)|}{|T (L)|} \frac{|T (J)|}{|T (R)|} ≥ \frac{|I|}{|L|} \frac{|J|}{|R|}.  (15)

This is a consequence of the Koebe Principle. It follows that |T̂′|^{− \frac{1}{2}} is convex on each branch and therefore \frac{d}{dx}|T̂′|^{− \frac{1}{2}} = −\frac{1}{2} T̂′′|T̂′|^{− \frac{3}{2}} is nondecreasing. Hence, each branch domain (a_{W} , b_{W}) =^{def} W contains at most one point r_{W} at which T̂′′ changes sign. We get

Var_{Z}g ≤ Var_{W} g
= ∫_{a_{W}}^{r_{W}} \frac{|T̂′′(s)|}{|T̂′(s)|^{1+t}} ds + ∫_{r_{W}}^{b_{W}} \frac{|T̂′′(s)|}{|T̂′(s)|^{1+t}} ds

≤ sup_{W} \frac{1}{|T̂′|^{t}} [|log \frac{T̂′(r_{W})}{T̂′(a_{W})}| + |log \frac{T̂′(b_{W})}{T̂′(r_{W})}|]
≤ 2K_{0}^{t} log K_{0} sup_{Z} g.

Therefore

VarX̂g ≤ ∑_{Z∈Z} Var_{Z}g + 2 ∑_{Z∈Z} sup_{Z} g
≤ ∑_{Z∈Z} 2 (K_{0}^{t} log K_{0} + 1) sup_{Z} g ≤ 2K_{0}^{t} (K_{0}^{t} log K_{0} + 1) < ∞.

Thus T̂ is a R-map and it satisfies the assumptions of Proposition 3.1. By Theorems 3.2 and 2.1, the conclusion follows because the set of X̂ satisfying the above assumptions have full measure.

In the proof of this theorem we even get that Z is a Markov partition for the induced map T̂ (in fact, each monotone branch is onto). Note that this is much better than what is necessary, since R-maps need not to satisfy this extremely strong topological property. This supports our belief that this method could be in principle applied to much more general systems, especially those with singularities.

4.2. Maps with neutral fixed points. Let α ∈ (0, 1) and consider the map T_{α} defined on X = [0, 1] by

T_{α}(x) = \{ \begin{matrix}x(1 + 2^{α}x^{α}) & if x ∈ [0, 1/2), \\ 2x − 1 & otherwise. \end{matrix}

Let μ_{α} denote the invariant measure absolutely continuous with respect to Lebesgue (see e.g. [26] for the existence and properties). The system (X, T_{α}, μ_{α}) has exponential return time statistics around cylinders of some naturally associated partition [20]; Here we prove that this is also true if the neighborhoods are balls.

Theorem 4.2. For any α ∈ (0, 1) the system ([0, 1], T_{α}, μ_{α}) has exponential return time statistic.

Proof. Let X̂_{0} =^{def} (\frac{1}{2}, 1] and for n ≥ 1

X̂_{n} =^{def} \{x ∈ X : T^{n}(x) > \frac{1}{2} and for k = 0, 1, …, n − 1, T^{k}(x) < \frac{1}{2}\}.

Fix n ≥ 0 and let X̂ = X̂_{n}. Let T̂ : X̂ → X̂ be the first return map to X̂. We then define a partition of X̂ by Z = {Z_{p} : p = 1, 2, … }, where Z_{p} = {x ∈ X̂ : τ̂X̂(x) = p}. One easily check that Z is a partition into intervals, and each branch of T̂ : Z → X̂ is monotone and onto. Let m be the normalized Lebesgue measure restricted to X̂ and let g(x) = 1/|T̂′(x)| when x ∈ int(Z) for some Z ∈ Z and g(x) = 0 otherwise. In the proof of Proposition 3.3 in [26] it is proved that for some constant K

sup_{x,y∈Z} \frac{T̂′(x)}{T̂′(y)} ≤ K

for all Z ∈ Z. A straightforward computation shows that

log K ≥ |log \frac{T̂′(x)}{T̂′(y)}| ≥ |∫_{x}^{y} \frac{T̂′′(t)}{T̂′(t)}dt|

for all x, y ∈ Z and Z ∈ Z. Since T̂ is increasing and convex on each Z ∈ Z, we get

∫_{Z} \frac{|T̂′′(t)|}{|T̂′(t)|^{2}} dt ≤ sup_{Z} g log K.

Taking into account that T̂Z = X̂ for any Z ∈ Z, we get

sup_{Z} g ≤ K \frac{m(Z)}{m(T̂(Z))} ≤ Km(Z).

Therefore, since g is C^{1} in the interior of Z and g|_{∂Z} = 0,

Var g ≤ ∑_{Z∈Z} ∫_{Z} |g′(t)|dt + 2 ∑_{Z∈Z} sup_{Z} g ≤ K(2 + log K) < ∞.

Finally, it is obvious that sup g < 1, hence (X̂, T̂, m) is an R-map. By Theorems 3.2 and 2.1 μ_{α}-almost all points inside X̂ have exponential return time statistics, and the conclusion follows since ∪_{n≥0}X̂_{n} has full Lebesgue measure.

5. Complex quadratic maps

In this section we apply the main framework to certain polynomials on the Riemann sphere ℂ̄. Every rational map T has a |T′|^{t}-conformal measure m_{t} for some t ∈ (0, 2], see Sullivan [43]. If T is hyperbolic on the Julia set J, then we can take t equal to the Hausdorff dimension of J, and m_{t} is equivalent to t-dimensional Hausdorff measure. In general however, m_{t} can be supported on a proper subset of J; it can even be atomic. This can be an issue if e.g. T has neutral periodic points.

For our results we assume that the orbit of the critical point does not densely fill the Julia set, but we do not require a supremum gap as in e.g. [16].

We start with a complex version of R-map and show (analogous to Theorem 3.2) that under some additional conditions they have exponential return time statistics.

Definition 5.1 (complex Markov maps). Let T : Y → X be a continuous map, Y ⊂ X open subsets of ℂ̄, and m_{t}(Y) = 1 where m_{t} is a probability measure and 0 < t ≤ 2. Let S = X \ Y. We call T a complex Markov map if the following is true:

1. There exists a countable family Z of pairwise disjoint open discs such that ∪_{Z∈Z}Z = Y;

2. For any Z ∈ Z, T : Z → X is a conformal diffeomorphism (with T (Z) = X), and with bounded distortion:

sup_{Z∈Z} sup_{Z} \frac{|T′′|}{|T′|^{2}} < ∞;  (16)

3. The measure m_{t} is |T′|^{t}-conformal;

4. The map T is expanding: inf_{y∈Y} |T′(y)| > 1.

5. Let Z_{k} = Z ∨ T^{−1}Z ∨ ⋯ ∨ T^{−(k−1)}Z is the k-th join of the partition Z. The domains Z ∈ Z_{k} are uniformly convex-like, by which we mean

sup_{k} sup_{Z∈Z_{k}} sup_{x≠y∈Z} \frac{p_{Z}(x, y)}{|x − y|} < ∞,  (17)

where p_{Z}(x, y) denotes infimum of the lengths of the path in Z connecting x and y.

Analogous to Theorem 3.1 one can show that complex Markov maps have an invariant measure with exponential decay of correlations.

Theorem 5.1. Let T be a complex Markov map as above. There exists an invariant ergodic probability measure μ equivalent to m_{t}. Moreover (X, T, μ) is mixing with exponential decay of correlations : There exists C > 0 and ρ ∈ (0, 1) such that for any f Lipschitz and g ∈ L_{μ}^{1}

|∫ f · g ∘ T^{n}dμ − ∫ fdμ ∫ gdμ| ≤ C∥f∥_{Lips}∥g∥_{L_{μ}^{1}}ρ^{n},  (18)

where ∥f∥_{Lips} = sup |f| + sup_{x≠y} \frac{|f(x)−f(y)|}{|x−y|}.

Proof. First let us make the following remark. As T is expanding, a straightforward calculation shows that (16) implies that

K := sup_{n∈ℕ} sup_{Z∈Z_{n}} sup_{w∈Z} \frac{|(T^{n})′′|}{|(T^{n})′|^{2}} < ∞

and ∑_{Z_{n}∈Z_{n}} sup_{Z} |(T^{n})′|^{−t} < ∞ uniformly in n. Let P_{t} be the Perron-Frobenius operator

(P_{t}f)(z) = ∑_{T (y)=z} \frac{f(y)}{|T′(y)|^{t}} .

Let us write ∥f∥_{s} for the seminorm sup_{z≠z′} \frac{|f(z)−f(z′)|}{|z−z′|}. If ∥f∥_{Lips} < ∞, we have

∥P_{t}^{n}f∥_{s} ≤ sup_{z,z′∈Y} \frac{1}{|z − z′|}|P_{t}^{n}(f)(z) − P_{t}^{n}(f)(z′)|

≤ sup_{z,z′∈Y} \frac{1}{|z − z′|} \{∑_{y} \frac{|f(y) − f(y′)|}{|(T^{n})′(y)|^{t}}+

∑_{y′} |f(y′)| |\frac{1}{|(T^{n})′(y)|^{t}} − \frac{1}{|(T^{n})′(y′)|^{t}} |\}

Here we summed over the pairs y, y′ in the same atom Z_{n} ∈ Z_{n} with T^{n}(y) = z and T^{n}(y′) = z′. As T : Z → X is onto for every Z ∈ Z, these pairs are well-defined. The first term in the above expression is bounded by

∑_{Z_{n}} sup_{y,y′∈Z_{n}} \frac{|f(y) − f(y′)|}{|y − y′|} \frac{1}{|(T^{n})′(y)|^{1+t}} ≤ ∥f∥_{s}∥P_{t}^{n}1I∥_{∞} sup_{y∈Y} \frac{1}{|(T^{n})′(y)|}.

Next we use the Mean Value Theorem and (17) to estimate

|\frac{1}{|(T^{n})′(y)|^{t}} − \frac{1}{|(T^{n})′(y′)|^{t}} | ≤ K̃ · (\frac{1}{|(T^{n})′(w)|^{t}} )′ |y − y′|
= \frac{K̃|t| |y − y′|}{|(T^{n})′(w)|^{t−1}} .\frac{|(T^{n})′′(w)|}{|(T^{n})′(w)|^{2}} ,

for some w. The constant K̃ is an upper bound in (17). This gives for the second term

sup_{Z∈Z} sup_{z,z′∈Z} ∑_{y′} \frac{f(y′)}{|z − z′|} |\frac{1}{|(T^{n})′(y)|^{t}} − \frac{1}{|(T^{n})′(y′)|^{t}} |

≤ ∥f∥_{∞} ∑_{Z_{n}} sup_{y,y′∈Z_{n}} \frac{1}{|y − y′| · |(T^{n})′(y)|} |\frac{1}{|(T^{n})′(y)|^{t}} − \frac{1}{|(T^{n})′(y′)|^{t}} |

≤ ∥f∥_{∞} ∑_{Z_{n}} sup_{w∈Z_{n}} \frac{K̃|t|}{|(T^{n})′(w)|^{t}} \frac{|(T^{n})′′(w)|}{|(T^{n})′(w)|^{2}}

≤ 2KK̃∥f∥_{∞}∥P_{t}^{n}1I∥_{∞}.

Let θ = sup |T′|^{−1} ∈ (0, 1). Then

\frac{∥P_{t}^{n}f∥_{s}}{∥P_{t}^{n}1I∥_{∞}} ≤ θ^{n}∥f∥_{s} + 2KK̃∥f∥_{∞}.

Obviously ∥P_{t}^{n}f∥_{∞} ≤ ∥f∥_{∞}∥P^{n}1I∥_{∞}. Therefore also

\frac{∥P_{t}^{n}f∥_{Lips}}{∥P_{t}^{n}1I∥_{∞}} ≤ θ^{n}∥f∥_{Lips} + (2KK̃ + 1)∥f∥_{∞}.

By construction ∥P_{t}^{n}1I∥_{∞} = ∑_{Z∈Z} sup_{Z} \frac{1}{|(T^{n})′|^{t}} < ∞ uniformly in n. This allows us to use the Tulcea-Ionescu & Marinescu theorem, which shows that P_{t} is a quasicompact operator. Since each branch is onto, 1 is a simple eigenvalue with Lipschitz eigenvector h > 0, and is the unique eigenvalue on the unit circle. Consequently, for any Lipschitz function f we have

∥P_{t}^{n}(fh) − h ∫fdμ∥_{Lips} ≤ C_{0}∥fh∥_{Lips}ρ^{n},

for some C_{0} > 0 and ρ ∈ (0, 1). It follows that the correlations between Lipschitz functions f and L_{μ}^{1} functions decays exponentially fast: if g ∈ L_{μ}^{1} we find

|∫ f · g ∘ T^{n}dμ − ∫fdμ ∫ gdμ| = |∫ (P_{t}^{n}(fh) − h ∫fdμ) · gdm_{t}|
≤ ∥P_{t}^{n}(fh) − h ∫fdμ∥_{∞} ∫ |g|dm_{t}
≤ C∥f∥_{Lips}∥g∥_{L_{μ}^{1}}ρ^{n},

for C = C_{0}∥h∥_{Lips}∥1/h∥_{∞}.

Theorem 5.2. Any complex Markov map T as defined above with |T′|^{t}- conformal measure m_{t} for some t > 1 admits an invariant mixing measure μ equivalent to m_{t} with exponential return time statistics around balls.

The proof of Theorem 5.2 is similar to that of Theorem 3.2. The additional difficulty is that the decay of correlation is given for Lipschitz function and not characteristic function of balls. The usual way to overcome this problem is to approximate balls by an union of small cylinders. This is the object of the next lemmas. Note that if we restrict ourselves to cylinder sets, then the theorem is valid for all t > 0.

As before, let Z_{k} = Z ∨ T^{−1}Z ∨ ⋯ ∨ T^{−k+1}Z denote the k-dynamical partition and B_{k} the σ-algebra generated by Z_{k}.

Lemma 5.1. Let T be a complex Markov map. There exists some constants Γ and γ > 0 such that for any set A ∈ B_{k} and Borel set B we have

|μ(A ∩ T^{−n−k}B) − μ(A)μ(B)| ≤ Γμ(A)μ(B) exp(−γn).  (19)

Proof. We can rewrite (19) as

|∫ h^{−1} · P_{t}^{k}(h1I_{A}) · 1I_{B} ∘ T^{n}dμ − μ(A)μ(B)| ≤ Γμ(A)μ(B) exp(−γn).

where h = \frac{dμ}{dm_{t}} is Lipschitz and bounded away from below (see [46]), and P_{t} is the Perron-Frobenius operator as defined in the previous proof. Recall

also that K =^{def} sup_{k∈ℕ} sup_{Z∈Z_{k}} sup_{Z} \frac{|(T^{k})′′|}{|(T^{k})′|^{2}} < ∞. Since T^{k} : A → X is

one-to-one and onto when A ∈ Z_{k}, the above facts imply that

sup_{k∈ℕ} sup_{A∈Z_{k}} \frac{∥h^{−1}P_{t}^{k}(h1I_{A})∥_{Lips}}{μ(A)} < ∞.

The lemma follows now from (18) in Theorem 5.1 by taking f = P_{t}^{k}(1I_{A})/μ(A) and g = μ(A)1I_{A}.

Given S ⊂ ℂ we denote by B_{k}(S) the smallest element of the σ-algebra B_{k} containing S ∩ J.

Lemma 5.2. There exists α > 0 such that for any z ∈ Y, r > 0 and k > 0 we have

μ (B_{k}(U_{r}(z)) ∩ B_{k}(U_{r}(z)^{c})) ≤ exp(−αk),  (20)

where U_{r}(z) denotes the ball of radius r about the point z.

Proof. Let us assume for simplicity that diam(Y) ≤ 1. We denote by int_{J}S the interior of a subset S ⊂ J in the induced topology. Since T is expanding, there exists a ∈ (0, 1) such that diam(Z) < a^{k} for any Z ∈ Z_{k} and integer k. By the Markov property, the bounded distortion and the conformality of the map T we can find some constant c > 0 such that the following property holds: For any integer k and cylinder Z ∈ Z_{k} there exists p_{Z} ∈ int_{J}Z and r_{Z} > c · diam(Z) such that U_{r_{Z}}(p_{Z}) ∩ int_{J}Z′ = ∅ for any Z ≠ Z′ ∈ Z_{k} different from Z. Given Z ≠ Z′ ∈ Z_{k}, we have d(p_{Z}, p_{Z′}) > max(r_{Z}, r_{Z′}); in particular, U_{r_{Z}/2}(p_{Z}) ∩ U_{r′_{Z}/2}(p_{Z′}) = ∅.

Let x ∈ Y, r > 0 and consider the following partition

P = {Z ∈ Z_{k} : Z ⊂ B_{k}(U_{r}(z))}.

Let P_{n} = {Z ∈ P : a^{n} < diam(Z) ≤ a^{n−1}}. Any Z ∈ P_{n} is a subset of the annulus S_{n}(z, r) =^{def} U_{r+a^{n}}(z) \ U_{r−a^{n}}(z). Thus there exists card(P_{n}) disjoint balls or radius at least a^{n}/2 inside S_{n}(z, r). Since the area of S_{n}(z, r) is equal to 4πra^{n} when a^{n} ≤ r we get

card(P_{n}) ≤ \frac{4πra^{n}}{c^{2}a^{2n}/4} = \frac{8π}{c^{2}} ra^{−n} ≤ \frac{8π}{c^{2}} a^{−n}.

Obviously when a^{n} > r we also have card(P_{n}) ≤ \frac{8π}{c^{2}} ≤ \frac{8π}{c^{2}} a^{−n}.

Since the measure m_{t} is |T′|^{t}-conformal and the map itself is conformal and Markov we have for some constant c_{1}, m_{t}(Z) < c_{1}diam(Z)^{t} for any Z ∈ Z_{k} for some k. The previous inequalities imply (observe that P_{n} = ∅ if n ≤ k and recall that t > 1)

μ( ⋃_{Z∈P} Z) = ∑_{n>k} μ( ⋃_{Z∈P_{n}} Z)

≤ ∑_{n>k} max{μ(Z) : Z ∈ P_{n}} · card(P_{n})

≤ ∑_{n>k} c_{1}a^{−t}a^{tn} \frac{8π}{c^{2}} a^{−n}

= \frac{8πc_{1}}{(a − a^{t})c^{2}} a^{(t−1)k}.

Taking α ∈ (0, (1 − t) log a) sufficiently small gives the result.

Proof of Theorem 5.2. The proof closely follows the one of Theorem 3.2, we use the same notation a_{N} and b_{N}. Let U = U_{r}(z) and k, N ∈ ℕ to be chosen later on. By Lemma 5.1 and Lemma 5.2

a_{N}(U) ≤ ∑_{n=τ(U)}^{N} \frac{1}{μ(U)}μ(B_{n}(U) ∩ T^{−n}U)

≤ (1 + Γ) ∑_{n=τ(U)}^{N} μ(B_{n}(U))

≤ \frac{1 + Γ}{1 − exp(−α)} [Nμ(U) + exp(−ατ(U))] .

Similarly, we get by Lemmas 5.1 and (twice) Lemma 5.2

b_{N}(U) ≤ Γμ(B_{k}(U)) exp[−γ(N − k)] + \frac{1}{μ(U)}μ(B_{k}(U) ∩ B_{k}(U^{c}))

≤ Γ exp[−γ(N − k)] + \frac{1}{μ(U)} (Γ exp[−αk − γ(N − k)] + exp[−αk]) ,

for all k ≤ N. Taking k = −\frac{2}{α} log μ(U) and N = 2k gives

b_{N}(U) ≤ Γμ(U)^{\frac{2γ}{α}} + Γμ(U)^{1+ \frac{2γ}{α}} + μ(U).

For all non periodic points z ∉ ∪_{k∈ℕ}∂Z_{k} we have τ(U_{r}(z)) → ∞, which implies c(U_{r}(z)) → 0 as r → 0. Since this concerns μ-almost all points in J, the theorem is proved.

We will apply these results to quadratic maps on ℂ. Induced systems have been used before for rational maps, notably by Aaronson et al. [1]. They consider parabolic rational maps (i.e. rational maps whose Julia sets contain no critical point but rationally indifferent periodic points), and establish the existence of an invariant measure μ ≪ m_{t}, where m_{t} is a t-conformal measure with t = HD(J). Moreover, μ is finite if and only if t · min_{p} \frac{a(p)+1}{a(p)} > 2. Here the minimum is taken over all parabolic points p and a(p) is such that T^{q}(z) = z + α(z − p)^{a(p)} + … for the appropriate iterate q. It follows that μ is finite only if t > 1, which is the hypothesis in Theorem 5.2. It is to be expected therefore that parabolic rational maps with a finite invariant measure μ ≪ m_{t} have exponential return time statistics on balls.

Theorem 5.3. Let T (z) = z^{2} + c be a quadratic map on ℂ such that T is not infinitely renormalizable (see discussion below) and its Julia set J contains no Cremer points. Suppose that also for some t > 1

• J supports a non-atomic |T′|^{t}-conformal measure m_{t},

• m_{t}(\overline{orb(Crit)}) = 0, and

• T preserves a probability measure μ equivalent to m_{t}.

Then (supp(μ), T, μ) has exponential return time statistics on disks.

The Hausdorff dimension of the Julia set HD(J) > 1 for each parameter c in the Mandelbrot set M (see Zdunik, [51]); the two exceptions c = 2 and c = 0 are easy real one-dimensional cases. So assuming that these parameter values allow a t-conformal measure with t = HD(J), the condition t > 1 is no restriction. By continuity, there are many parameters close to M that give t > 1 as well.

Note that we allow T to have parabolic points or Siegel disks. The map T is called renormalizable if there exist open disks W_{0} and W_{1}, 0 ∈ W_{0} ⊂ W_{1}, such that T^{n} : W_{0} → W_{1} is a two-fold covering map for some n ≥ 2 and T^{in}(0) ∈ W_{1} for all i ≥ 1. If there are infinitely many integers n such that this is possible, T is infinitely renormalizable. We assume that T is not infinitely renormalizable and has no Cremer point to be able to use Yoccoz’ puzzle construction. In particular the result that for each z ∈ J, the puzzle pieces P_{n}(z) containing z shrink to z as n → ∞ is important. For an exposition of Yoccoz’ puzzles and the proof of these statements we refer to [31].

Yoccoz’ results have been extended to certain infinitely renormalizable polynomials by Lyubich and Levin & van Strien, [27, 24]. For reasons of simplicity, we have not tried to extend Theorem 5.3 to these cases; we prefer to work with a single set of initial puzzle pieces P_{0}. Prado [37] has shown that in all of the above cases, the conformal measure m_{t} is ergodic.

Proof. Let z ∈ supp(μ) \ \overline{orb(Crit)} be arbitrary, and let U ⊃ V ∋ z be open disks such that U ∩ \overline{orb(Crit)} = ∅ and V is compactly contained in U. Let log δ = mod(U \ V) > 0 be the modulus of U \ V. Assume also that V is convex-like in the sense that sup_{x≠y∈V} p_{V} (x, y)/|x − y| < ∞, where p_{V} is as in (17).

The strategy is to find a subset X̂ of V such that T^{n}(∂X̂) ∩ X̂ = ∅ for all n ≥ 0, and then we can invoke Theorem 5.2.

If J is a Cantor set, we can assume that ∂V is contained in the Fatou set F, which is the basin of ∞ in this case. Moreover, there are no neutral or stable periodic orbits. Thus each point in F, in particular ∂V, converges to ∞. It follows that T^{n}(∂V) intersects V for at most finitely many n ≥ 0. Let X̂ be the component of V \ ∪_{n≥0}T^{n}(∂V) containing z. Then T^{n}(∂X̂) ∩ X̂ = ∅ for all n ≥ 0 as required. Since X̂ consists of the intersection and difference of at most finitely many, convex-like disks, we find sup_{x≠y∈X̂} pX̂(x, y)/|x−y| ≤ C(X̂) < ∞.

Assume now that J is connected and by Yoccoz’ results also locally connected. Let F_{i}, i ≥ 0, be the periodic components of the Fatou set, with F_{0} the basin of ∞. Since T is a polynomial (with exceptional point ∞), J = ∂F_{0}. There are only finitely many such components, and by Sullivan’s Theorem (see e.g. [44]), every z′ ∈ F is eventually mapped into ∪_{i}F_{i}. We construct a special forward invariant subset G of \overline{∪_{i}F_{i}}.

1. Consider the renormalization T^{n} : W_{0} → W_{1} of the highest possible period n. (If T is not renormalizable, then we just take T : ℂ → ℂ.) It is known that W_{1} contains an n-periodic point p with at least two external rays, say A_{0} and A′_{0}. The existence of such external rays (and the arcs A_{i} defined later on) is guaranteed by results initiated by Douady, see [35] and references therein.) Let G_{0} = {z ∈ ℂ; |z| > 10}.

2. If F_{i}, i ≥ 1, contains a stable periodic point, let G_{i} be a disk compactly contained in F_{i} such that T^{per(F_{i})}(G_{i}) ⊂ G_{i} and orb(Crit ∩ F_{i}) ⊂ G_{i}. There is at least one per(F_{i})-periodic point p_{i} in the boundary of F_{i}. Let A_{i} be a smooth compact arc connecting p_{i} and G_{i} such that T^{per(F_{i})}(A_{i}) ⊂ A_{i} ∪ G_{i}. Since p_{i} ∈ ∂F_{0}, there is also an external ray A′_{i} landing at p_{i}.

3. If F_{i} contains a parabolic point p_{i} in its boundary such that each z ∈ F_{i} converges to p_{i}, let G_{i} ⊂ F_{i} be a disk such that T^{per(F_{i})}(G_{i}) ⊂ G_{i}, orb(Crit) ∩ F_{i} ⊂ G_{i}, and ∂G_{i} ∩ ∂F_{i} = {p_{i}}. Let A′_{i} ⊂ F_{0} be an external ray landing at p_{i}.

Let

G = ⋃_{i≥0}(G_{i} ∪ {p_{i}} ∪ A_{i} ∪ A′_{i}).

Then G is connected, and for some N ∈ ℕ, ∪_{j=1}^{N}T^{j}G is forward invariant. We start a Yoccoz puzzle construction by putting, for n ≥ 0,

P_{n} = {components of ℂ̄ \ T^{−n}G}

For each n ≥ 1, T maps any element of of P_{n} into an element of P_{n−1}. Using the arguments in [31], one can show that the diameters of the elements Y ∈ P_{i} tend to 0 as i → ∞, unless Y eventually intersects a Siegel disk.

We can assume that the point z ∉ ∪_{n}T^{−n}(G). Moreover, since orb(Crit) densely fills the boundary of any Siegel disk, and m_{t}(\overline{orb(Crit)}) = 0, we can assume that z does not lie on the boundary of a Siegel disk. Find n so large that the element Y of P_{n} containing z is contained in V. Let X̂ = V ∩ Y. Then T^{n}(∂X̂) ∩ X̂ = ∅ for all n ≥ 0. Note also that T^{n}(Y) is bounded by finitely many smooth curves of ∂G_{i}, A_{i} and A′_{i}. At worst these curves end in a logarithmic spiral, namely as they approach the hyperbolic periodic points p_{i}. Therefore also T^{n}(X̂) is convex-like, and obviously simply connected. It follows that sup_{x≠y∈X̂} pX̂(x, y)/|x − y| ≤ C(X̂) < ∞.

The rest of the argument works for both J locally connected and J a Cantor set. Let T̂ : X̂ → X̂ be the first return map to X̂. Then T̂ is defined on a countable collection Z of disks Z. The modulus mod(U \ X̂) ≥ log δ, and for each branch T̂ = T^{τ} : Z → X̂ there exists a disk Z′ ⊃ Z such that T^{τ} maps Z′ univalently onto U. It follows from the Koebe \frac{1}{4}-Theorem (e.g. [8, Theorem 1.4]) that the distortion of T̂|_{Z} is uniformly bounded:

sup_{x,y∈Z} \frac{|T̂′(x)|}{|T̂′(y)|} ≤ K = K(δ)  (21)

More precisely, take x ∈ Z and let [x, y] ⊂ Z be a straight arc containing x such that T̂′′/T̂′ varies little on A. Then

\frac{|T̂′′(x)|}{|T̂′(x)|} ≤ \frac{2}{|y − x|} |∫_{x}^{y} \frac{T̂′′(u)}{T̂(u)}du| ≤ \frac{2}{|y − x|} log \frac{|T̂′(y)|}{|T̂(x)|}.

To estimate this, let B be the maximal round disk centered at x contained in Z′. Let δ_{0} be the radius of B; we have δ_{0}|T̂′(x)| = O(δ). Define f(w) = \frac{T̂(x+δ_{0}w)−T̂(x)}{δ_{0}T̂′(x)}. Then f is a univalent map on the unit disk with f′(0) = 1. By Theorem 1.6. of [8], we obtain

\frac{|T̂′(y)|}{|T̂′(x)|^{2}} = \frac{1}{|T̂′(x)|} log \frac{|f′(\frac{|y−x|}{δ_{0}})|}{|f′(0)|}

≤ \frac{1}{|T̂′(x)|} [log(1 + \frac{|y − x|}{δ_{0}}) − 3 log(1 − \frac{|y − x|}{δ_{0}})]

≤ \frac{5|y − x|}{δ_{0}|T̂′(x)|},

provided y is close to x. Therefore \frac{|T̂′′(x)|}{|T̂′(x)|^{2}} ≤ \frac{10}{δ_{0}|T̂′(x)|}, proving that

sup_{Z∈Z} sup_{Z} \frac{|T̂′′|}{|T̂′|^{2}} < ∞.  (22)

The distortion bound K(δ) applies by the same argument also to iterates T̂^{n} of the induced map. Therefore each domain Z ∈ Z_{n} is not much less convex than X̂:

sup_{k} sup_{Z∈Z_{k}} sup_{x≠y∈Z} \frac{p_{Z}(x, y)}{|x − y|} ≤ C

for some C = C(δ, X̂) < ∞.

We can also assume that V and hence X̂ is so small that inf_{∪Z} |T̂′(x)| > 1. Thus T̂ is hyperbolic. Recall that m_{t} is the |T′|^{t}-conformal measure of T. As T̂ is a first return map, it is straightforward to show that m̂_{t} = \frac{m_{t}}{m_{t}(X̂)}

is a |T̂′|^{t}-conformal map for T̂. In fact, m̂_{t} can also be constructed using Sullivan’s techniques [43]. Now we can invoke Theorem 5.2 with the invariant measure μ̂ = \frac{1}{μ(X̂)}μ.

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Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, the Netherlands

E-mail address: bruin@math.rug.nl
URL: http://www.math.rug.nl/~bruin

Laboratoire Amiénois de Mathémathiques Fondamentales et Appliquées, Université de Picardie–Jules Verne, 33 rue St Leu, 80039 Amiens Cedex 1, France

E-mail address: benoit.saussol@mathinfo.u-picardie.fr
URL: http://www.mathinfo.u-picardie.fr/saussol/

Centre de Physique Théorique et Institut de Mathématiques de Luminy, FRU-MAM, Féderation de Recherche des Unites de Mathematiques de Marseille, CNRS Luminy, Case 907, F-13288 Marseille Cedex 9, France

E-mail address: serge@cpt.univ-mrs.fr
E-mail address: troubetz@iml.univ-mrs.fr
URL: http://iml.univ-mrs.fr/~troubetz/

Centre de Physique Théorique, FRUMAM, Féderation de Recherche des Unites de Mathematiques de Marseille, CNRS Luminy, Case 907, F-13288 Marseille Cedex 9, France

E-mail address: vaienti@cpt.univ-mrs.fr
URL: http://www.cpt.univ-mrs.fr/