Non-hyperbolic behavior of geodesic flows of rank 1 surfaces

Katrin Gelfert

doi:10.3934/dcds.2019022

Article Contents

2019, Volume 39, Issue 1: 521-551. Doi: 10.3934/dcds.2019022

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Non-hyperbolic behavior of geodesic flows of rank 1 surfaces

Katrin Gelfert

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Av. Athos da Silveira Ramos 149, Rio de Janeiro 21945-909, Brazil

KG has been supported by CNPq (Brazil). She is very grateful for the comments by the referee

Received: March 31, 2018

Revised: June 30, 2018

Published: December 2018

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Abstract

We prove that for the geodesic flow of a rank 1 Riemannian surface which is expansive but not Anosov the Hausdorff dimension of the set of vectors with only zero Lyapunov exponents is large.

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Mathematics Subject Classification: Primary: 53D25, 37D40, 37D25, 37D35, 37C45.

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Figure 1. local product structure

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Figure 2. Parametrization $R = R_v$ of the local cross section in a neighborhood of a vector $v$. Here $\pi$ denotes the projection of the centre stable leaf onto the cross section given by the local product structure.

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Figure 3. Schematic construction of $\nu$: $m_\ell$-cylinders which intersect $\Sigma^\ell$ (bold), cylinders on which $\nu$ is distributed (bold blue), $\ell = 1,2,3$

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