Induced maps of hyperbolic Bernoulli systems
Matthew Nicol
Discrete & Continuous Dynamical Systems - A 2001, 7(1): 147-154 doi: 10.3934/dcds.2001.7.147
Let $(f,T^n,\mu)$ be a linear hyperbolic automorphism of the $n$-torus. We show that if $A\subset T^n$ has a boundary which is a finite union of $C^1$ submanifolds which have no tangents in the stable ($E^s$) or unstable $(E^u)$ direction then the induced map on $A$, $(f_A,A,\mu_A)$ is also Bernoulli. We show that Poincáre maps for uniformly transverse $C^1$ Poincáre sections in smooth Bernoulli Anosov flows preserving a volume measure are Bernoulli if they are also transverse to the strongly stable and strongly unstable foliation.
keywords: Induced map Bernoulli dynamical system.
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
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.

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