Anosov diffeomorphism with a horseshoe that attracts almost any point

Christian Bonatti; Stanislav Minkov; Alexey Okunev; Ivan Shilin

doi:10.3934/dcds.2020017

Article Contents

2020, Volume 40, Issue 1: 441-465. Doi: 10.3934/dcds.2020017

This issue Previous Article On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations Next Article Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type

Anosov diffeomorphism with a horseshoe that attracts almost any point

1.
Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France
2.
Brook Institute of Electronic Control Machines, 119334, Moscow, Vavilova str., 24, Russia
3.
Loughborough University, LE11 3TU, Loughborough, UK
4.
National Research University Higher School of Economics, Faculty of Mathematics, 119048, Moscow, Usacheva str., 6, Russia

Received: February 28, 2019

Revised: May 31, 2019

Published: October 2019

Supported in part by the RFBR grant 16-01-00748-a

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Abstract

We present an example of a $ C^1 $ Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.

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Mathematics Subject Classification: Primary: 37D20; Secondary: 37C40, 37C70, 37E30, 37G35.

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Figure 1. The set $ UK $. The vertical stripes of $ UK \setminus K $ are depicted wider than they are.

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Figure 2. The "frame" $ R $.

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Figure 3. $ \tilde R $, $ UH_i $, $ \tilde RH_i $, $ RH_i $.

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