First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

Abstract

We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of uniform dual ergodicity for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate.In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series, which to our knowledge, has not previously been covered.

Mixing rates for intermittent maps of high exponent

“…Higher order expansions for 1 − λ(z), z ∈ D (and thus for T (z), z ∈ D) were obtained in [24,27]. The assumptions in [24,27] are much more modest than the ones used in this work.…”

“…As shown in [24], much weaker versions of (H), (H1) and (H2) above are enough for first order expansion of (1 − λ(z)) −1 , and consequently of T (z), for z ∈ D, as z → 1. We recall this result as relevant to our setting.…”

“…Under more mild assumptions, the asymptotics of 1 − Ψ(z), z ∈ D was (implicitly) obtained in [24,27] in the process of understanding the asymptotics of 1 − λ(z), z ∈ D. First order expansion of 1 − Ψ(e iθ ) under the assumption µ(ϕ > n) = (n)n −β was obtained in several other works (see, for instance, [14]). …”

“…We need some functional-analytic assumptions on the first return map F : Y → Y . Our assumption (H1) below is stronger than assumption (H1) in [23,24,27]; it is of the same strength as the one in [18]. We assume that there is a function space B ⊂ L ∞ (Y ) containing constant functions, with norm satisfying |v| ∞ ≤ v for v ∈ B, such that:…”

“…The assumptions in [24,27] are much more modest than the ones used in this work. For higher order expansions of 1 − λ(e iθ ) under very mild assumptions, we refer the reader to [23].…”