Pinching conditions, linearization and regularity of Axiom A flows

Luchezar Stoyanov

doi:10.3934/dcds.2013.33.391

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2013, Volume 33, Issue 2: 391-412. Doi: 10.3934/dcds.2013.33.391

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Pinching conditions, linearization and regularity of Axiom A flows

Luchezar Stoyanov^1,

1.
School of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009

Received: May 31, 2011

Revised: April 30, 2012

Published: January 2013

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Abstract

In this paper we study a certain regularity property of $C^2$ Axiom A flows $\phi_t$ over basic sets $Λ$ related to diameters of balls in Bowen's metric, which we call regular distortion along unstable manifolds. The motivation to investigate the latter comes from the study of spectral properties of Ruelle transfer operators in [21]. We prove that if the bottom of the spectrum of $d\phi_t$ over $E^u_{|Λ}$ is point-wisely pinched and integrable, then the flow has regular distortion along unstable manifolds over $Λ$. In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate to its linearization over the `pinched part' of the unstable tangent bundle.

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Mathematics Subject Classification: Primary: 37D20; Secondary: 37A25.

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