Dimension theory of flows: A survey

Luis Barreira

doi:10.3934/dcdsb.2015.20.3345

Article Contents

2015, Volume 20, Issue 10: 3345-3362. Doi: 10.3934/dcdsb.2015.20.3345

This issue Previous Article On entropy, entropy-like quantities, and applications Next Article Brief survey on the topological entropy

Dimension theory of flows: A survey

Luis Barreira^1,

1.
Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa

Received: October 31, 2014

Revised: February 28, 2015

Published: November 2015

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Abstract

This is a survey on recent developments of the dimension theory of flows, with emphasis on hyperbolic flows. In particular, we describe various results of the dimension theory and multifractal analysis of flows, including the dimension of hyperbolic sets, the dimension of invariant measures, the multifractal analysis of equilibrium measures, conditional variational principles, multidimensional spectra, and dimension spectra taking both into account past and future. The dimension theory and the multifractal analysis of dynamical systems grew out exponentially during the last two decades, but for various reasons flows have initially been given less attention than maps. We emphasize that this is not the case anymore and the survey is also an invitation to the theory.

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Mathematics Subject Classification: Primary: 37C45, 37D20, 37D35.

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