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
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
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.
On a semigroup problem
Viorel Nitica Andrei Török
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 2365-2377 doi: 10.3934/dcdss.2019148

If $ S $ is a semigroup in $ \mathbb{R}^n $ that is not separated by a linear functional, then it is known that the closure of $ S $ is a group. We investigate a similar statement in an infinite dimensional topological vector space $ X $. We show that if $ X $ is an infinite dimensional Banach space, then there exists a semigroup $ S\subset X $, not separated by the continuous functionals supported by the closed linear span of $ S $, for which the closure of the semigroup is not a group. If $ X $ is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if $ X $ is $ \mathbb{R}^{\omega} $, the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as $ \mathbb{R}^{\infty} $, the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.

keywords: Semigroup group Banach space Fréchet space separation by functionals topological transitivity extension hyperbolic dynamical system infinite direct product of lines infinite direct sum of lines
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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