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DCDS
Statistical properties of compact group extensions of hyperbolic flows and their time one maps
Michael Field,  Ian Melbourne,  Matthew Nicol,  Andrei Török
Discrete & Continuous Dynamical Systems - A 2005, 12(1): 79-96 doi: 10.3934/dcds.2005.12.79
Recent work of Dolgopyat shows that "typical" hyperbolic flows exhibit rapid decay of correlations. Melbourne and Török used this result to derive statistical limit laws such as the central limit theorem and the almost sure invariance principle for the time-one map of such flows.
In this paper, we extend these results to equivariant observations on compact group extensions of hyperbolic flows and their time one maps.
keywords: decay of correlations ,  almost sure invariance principle ,  hyperbolic flow ,  axiom A. ,  Compact group extensions
JMD
Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure
Ian Melbourne,  Dalia Terhesiu
Journal of Modern Dynamics 2018, 12(1): 285-313 doi: 10.3934/jmd.2018011

We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgo-pyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly non-Markovian) nonuniformly expanding maps hold also for their toral extensions.
keywords: Toral extensions ,  decay of correlations ,  infinite ergodic theory
DCDS
A note about stable transitivity of noncompact extensions of hyperbolic systems
Ian Melbourne,  V. Niţicâ,  Andrei Török
Discrete & Continuous Dynamical Systems - A 2006, 14(2): 355-363 doi: 10.3934/dcds.2006.14.355
Let $f:X\to X$ be the restriction to a hyperbolic basic set of a smooth diffeomorphism. If $G$ is the special Euclidean group $SE(2)$ we show that in the set of $C^2$ $G$-extensions of $f$ there exists an open and dense subset of stably transitive transformations. If $G=K\times \mathbb R^n$, where $K$ is a compact connected Lie group, we show that an open and dense set of $C^2$ $G$-extensions satisfying a certain separation condition are transitive. The separation condition is necessary.
keywords: skew-product ,  stable transitivity ,  hyperbolic basic set. ,  Noncompact group extension

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