J. London Math. Soc. 70 (2004) 427-446.
Ian Melbourne and Matthew Nicol
Abstract
We consider the statistical properties of endomorphisms under the assumption that the associated Perron-Frobenius operator is quasicompact. In particular we consider the central limit theorem, weak invariance principle, law of the iterated logarithm and exponential decay of correlations for sufficiently regular observations. Our approach clarifies the role of the usual assumptions of ergodicity, weak-mixing and exactness.
We also give sufficient conditions for quasicompactness of the Perron-Frobenius operator to lift to the corresponding equivariant operator on a compact group extension of the base. This leads to statistical limit theorems and exponential decay of correlations for equivariant observations on compact group extensions.
Examples considered include compact group extensions of
piecewise uniformly expanding maps (for example Lasota-Yorke maps),
and subshifts of finite type, as well as systems that are nonuniformly
expanding or nonuniformly hyperbolic.