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Statistical properties of mostly contracting fast-slow partially hyperbolic systems

Research paper by Jacopo De Simoi, Carlangelo Liverani

Indexed on: 22 Oct '16Published on: 01 Oct '16Published in: Inventiones mathematicae



Abstract

Abstract We consider a class of \({\mathcal C}^{4}\) partially hyperbolic systems on \({\mathbb T}^2\) described by maps \(F_\varepsilon (x,\theta )=(f(x,\theta ),\theta +\varepsilon \omega (x,\theta ))\) where \(f(\cdot ,\theta )\) are expanding maps of the circle. For sufficiently small \(\varepsilon \) and \(\omega \) generic in an open set, we precisely classify the SRB measures for \(F_\varepsilon \) and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.AbstractWe consider a class of \({\mathcal C}^{4}\) partially hyperbolic systems on \({\mathbb T}^2\) described by maps \(F_\varepsilon (x,\theta )=(f(x,\theta ),\theta +\varepsilon \omega (x,\theta ))\) where \(f(\cdot ,\theta )\) are expanding maps of the circle. For sufficiently small \(\varepsilon \) and \(\omega \) generic in an open set, we precisely classify the SRB measures for \(F_\varepsilon \) and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate. \({\mathcal C}^{4}\) \({\mathcal C}^{4}\) \({\mathbb T}^2\) \({\mathbb T}^2\) \(F_\varepsilon (x,\theta )=(f(x,\theta ),\theta +\varepsilon \omega (x,\theta ))\) \(F_\varepsilon (x,\theta )=(f(x,\theta ),\theta +\varepsilon \omega (x,\theta ))\) \(f(\cdot ,\theta )\) \(f(\cdot ,\theta )\) \(\varepsilon \) \(\varepsilon \) \(\omega \) \(\omega \) \(F_\varepsilon \) \(F_\varepsilon \)