Operator renewal theory and mixing rates for dynamical systems with infinite measure

Abstract

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates L n of the transfer operator. This was previously an intractable problem.Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points.In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of n j=1 L j ) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for L n and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.This version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published version.

Mixing rates for intermittent maps of high exponent

“…As shown in [23], the above results on T n extend to similar results on L n associated with a large class of systems preserving an infinite measure. We recall that previous to the results in [23] via operator renewal techniques, Thaler [31] obtained first order asymptotics of L n for a rather restrictive class of dynamical systems, which applies to reasonably large classes of systems (similar to the family of maps (1.2) recalled in Section 1.3) just in the case β = 1.…”

“…The only other previous results on higher order asymptotic for scalar renewal sequences are contained in [23,27] and do not address the regime β ∈ (0, 1/2]. Also, we note that Theorem 1.1 improves the error terms in [23,27] for the range β ∈ (1/2, 1).…”

“…In the same work, the authors also obtain an optimal version of item i) above for the case β = 1. Since the case β = 1 has completely treated in [23] (also in the sense of higher order theory as explained below), we do not consider this case in the present work. For a different technique for operator renewal sequences satisfying the general assumption µ(ϕ > n) = (n)n −β (and implicitly, for scalar renewal sequences) we refer to [27].…”

“…In [23], the results summarized above are referred to as first order asymptotics of T n . In the same work, the authors also obtain an optimal version of item i) above for the case β = 1.…”

“…In [23], Melbourne and Terhesiu developed a theory of operator renewal sequences for dynamical systems with infinite measure, generalizing the results of [14,11] to the operator case. Under suitable assumptions on the first return map f ϕ , [23] shows that for a ('sufficiently regular') function v supported on Y and a constant d 0 = 1 π sin βπ, the following hold: i) when β ∈ (…”