Central Limit Theorems and Suppression of Anomalous Diffusion for Systems with Symmetry

Nonlinearity 29 (2016) 2941-2960.

Georg Gottwald and Ian Melbourne

Abstract

We give general conditions for the central limit theorem and weak convergence to Brownian motion (the weak invariance principle / functional central limit theorem) to hold for observables of compact group extensions of nonuniformly expanding maps. In particular, our results include situations where the central limit theorem would fail, and anomalous behaviour would prevail, if the compact group were not present. This has important consequences for systems with noncompact Euclidean symmetry.

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Typos, etc: For the CLT in Theorem 3.3, an extra assumption is required,namely n-1/2maxj<nΦo Fj->p0 (In the application, Φ∈ L2 so this extra condition is automatic.)