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
DCDS
Rigidity of partially hyperbolic actions of property (T) groups
Andrei Török
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 193-208 doi: 10.3934/dcds.2003.9.193
We show that volume-preserving perturbations of some product actions of property (T) groups exhibit a "foliation rigidity" property, which reduces the partially hyperbolic action to a family of hyperbolic actions. This is used to show that certain partially hyperbolic actions are locally rigid.
keywords: Kazdahn's property (T) invariant volume partially hyperbolic actions ergodicity local rigidity higher rank lattices. invariant foliations
DCDS
Polynomial loss of memory for maps of the interval with a neutral fixed point
Romain Aimino Huyi Hu Matthew Nicol Andrei Török Sandro Vaienti
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 793-806 doi: 10.3934/dcds.2015.35.793
We give an example of a sequential dynamical system consisting of intermittent-type maps which exhibits loss of memory with a polynomial rate of decay. A uniform bound holds for the upper rate of memory loss. The maps may be chosen in any sequence, and the bound holds for all compositions.
keywords: non-stationary dynamics sequential systems distortion loss of memory neutral fixed point Intermittency polynomial decorrelation.
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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