Statistical properties of one-dimensional expanding maps with singularities of low regularity

Jianyu Chen; Hong-Kun Zhang

doi:10.3934/dcds.2019203

Article Contents

2019, Volume 39, Issue 9: 4955-4977. Doi: 10.3934/dcds.2019203

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Statistical properties of one-dimensional expanding maps with singularities of low regularity

Jianyu Chen and
Hong-Kun Zhang

Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA

Received: February 28, 2018

Revised: December 31, 2018

Published: May 2019

The second author is partially supported by the NSF Career Award (DMS-1151762)

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Abstract

We investigate the statistical properties of piecewise expanding maps on the unit interval, whose inverse Jacobian may have low regularity near singularities. The method is new yet simple: instead of directly working with the 1-d map, we first lift the 1-d expanding map to a hyperbolic map on the unit square, and then take advantage of the functional analytic method developed by Demers and Zhang in [21,22,23] for hyperbolic systems with singularities. By projecting back to the 1-d map, we are able to prove that it inherits nice statistical properties, including the large deviation principle, the exponential decay of correlations, as well as the almost sure invariance principle for the expanding map on a large class of observables. Moreover, we are able to prove that the projected SRB measure has a piecewise continuous density function. Our results apply to rather general 1-d expanding maps, including some $ C^1 $ perturbations of the Lorenz-like map and the Gauss map whose statistical properties are still unknown as they fail all other available methods.

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Mathematics Subject Classification: Primary: 37D50, 37A25.

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Figure 1. Lorenz-like Map

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Figure 2. Gauss Map

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Figure 3. The two dimensional lifting map $ \widehat{F}: Q\backslash \widehat{\mathcal{S}}_1\to Q\backslash \widehat{\mathcal{S}}_{-1} $

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