Mixing rates and limit theorems for random intermittent maps

Abstract

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_{\alpha }$ using the full parameter range $0<\alpha <\infty$, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest $\alpha$) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the \emph{quenched} dynamics. The main applications of this estimate are to \textit{limit laws} (in particular, CLT and stable laws, depending on the parameters chosen in the range $0<\alpha <1$) for the associated skew product; these are detailed in Theorem 3.2. Since our estimates in Theorem 1.1 also hold for $1\le \alpha <\infty$ we study a piecewise affine version of our random transformations, prove existence of an infinite ($\sigma -$finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.

After publication, a silly mistake was found in definition of the skew product (4.1). The updated version includes a correction of equation (4.1) in the published version. All results of the paper are unaffected. In particular, everything in sections 1-3 and the proof in section 4 are unchanged from the published version

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• A spectral approach for quenched limit theorems for random expanding dynamical systems
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• Loss of memory and moment bounds for nonstationary intermittent dynamical systems
• Quasistatic dynamics with intermittency
• Critical intermittency in random interval maps
• Central limit theorems with a rate of convergence for time-dependent intermittent maps
• Decay of correlations for critically intermittent systems
• Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges
• Deviations from Gaussianity in deterministic discrete time dynamical systems