Renewal theorems and mixing for non Markov flows with infinite measure

Abstract

We obtain results on mixing for a large class of (not necessarily Markov) infinite measure semiflows and flows. Erickson proved, amongst other things, a strong renewal theorem in the corresponding i.i.d. setting. Using operator renewal theory, we extend Erickson's methods to the deterministic (i.e. noni.i.d.) continuous time setting and obtain results on mixing as a consequence.Our results apply to intermittent semiflows and flows of Pomeau-Manneville type (both Markov and nonMarkov), and to semiflows and flows over Collet-Eckmann maps with nonintegrable roof function.Here, ℓ : [0, ∞) → [0, ∞) is a measurable slowly varying function (so lim t→∞ ℓ(λt)/ℓ(t) = 1 for all λ > 0). Consider the suspension (Y τ , µ τ ) and suspension semiflow F t : Y τ → Y τ (the standard definitions are recalled in Section 3).The aim is to prove a mixing result of the formfor a suitable normalisation a t → ∞ and suitable classes of observables v, w :Under certain hypotheses, [13,37] obtained results on mixing and rates of mixing for such semiflows. The hypotheses were of two types: (i) assumptions on "renewal operators" associated to the transfer operator of F and the roof function τ , and (ii) Dolgopyat-type assumptions of the type used to obtain mixing rates for finite measure (semi)flows [17].As pointed out to us by Dima Dolgopyat, Péter Nándori and Doma Szász, mixing for indicator functions can be regarded as a local limit theorem and hence hypotheses of type (ii) should not be necessary.In this paper, we show that operator renewal-theoretic assumptions (i) are indeed sufficient for obtaining the mixing results in [13,37]. The abstract framework in [13] turns out again to be flexible enough to cover nonMarkov situations. Moreover, our main results extend to flows and we are able to treat large classes of observables v, w. (Conditions of type (i) alone are not sufficient for obtaining rates of mixing; the best results remain those in [13].)The analogous probabilistic results go back to Erickson [20] who obtained strong renewal theorems in an i.i.d. continuous time framework under the assumption β ∈ ( 1 2 , 1]. (In the discrete time setting, see [22] for the i.i.d. case and [35] for the deterministic case.) Our results on mixing when β ∈ ( 1 2 , 1] for semiflows (Corollary 3.1 and the extensions in Section 9) and for flows (Theorem 10.5), are proved by adapting Erickson's methods to the deterministic setting.For β ≤ 1 2 , additional hypotheses are needed on the tail of τ to obtain a strong renewal theorem (and hence mixing) even for discrete time; see [15,19,22] for i.i.d. results and [24] for deterministic results (see also [41] for higher order theory in both the i.i.d. and deterministic settings). For the continuous time case, Dolgopyat & Nándori [18] obtain strong renewal theorems for a class of Markov semiflows including the range β ≤ 1 2 (again under extra hypotheses on the tail µ(τ > t)), though our main examples seem beyond their framework. In the absence of additional tail hypotheses, [20] showed how to obtain a partial resul...

Infinite measure renewal theorem and related results

“…This notion of mixing is related to classical renewal theory [14] and to limit distributions of ergodic sums of infinite measure preserving transformations [10]. Recently, there was a considerable interest in studying mixing properties of hyperbolic transformations preserving an infinite measure in both discrete and continuous time settings (see [2,5,16,23,24,[26][27][28]31], and references wherein).…”

Local large deviations and the strong renewal theorem

“…We also obtain very explicit and sharp sufficient conditions, which refine those in the literature, see Propositions 1.7 and 1.17. Our results are referred to in the recent papers [Ber17, Ber18, Chi18, DN17, DW18, FMMV18, Kev17, Kol17, MT17,Uch18].…”

Analytic proof of multivariate stable local large deviations and application to deterministic dynamical systems

“…Remark 3.1. In [1,2,15], local limit theorems are proved in various situations under additional aperiodicity assumptions. Our results do not require aperiodicity assumptions, so we do not discuss these issues further.…”