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Exponential Mixing: Lectures from Mumbai

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Geometric and Ergodic Aspects of Group Actions

Part of the book series: Infosys Science Foundation Series ((ISFM))

Abstract

We discuss a number of results related to mixing and, in particular, to the rate of mixing. This is sometimes alternatively known as the rate of decay of correlations.

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Notes

  1. 1.

    This doesn’t have finitely many branches but the same basic analysis applies.

  2. 2.

    Actually, one needs a slightly weaker assumption: There exists \(\varepsilon > 0\) and (an arbitrary large) n such that \(T^n :X \rightarrow X\) has inverse branches \(T_i, T_j :X \rightarrow X\) (i.e. locally \(T^n \circ T_i =\) identity and \(T^n \circ T_j =\) identity) and \(R_{ij} := r(T_i x) - r(T_jx)\) satisfies \(|R^\prime _{ij}(x)| \ge \varepsilon \).

  3. 3.

    In the detailed proof there remains a complicated technical argument to get bounds of iterates of the operator and thus the spectral radius.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UK

    Mark Pollicott

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  1. Mark Pollicott
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Correspondence to Mark Pollicott .

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Editors and Affiliations

  1. UM-DAE, Center for Excellence in Basic Sciences, Mumbai, Maharashtra, India

    S. G. Dani

  2. School of Mathematics, Tata Institute of Fundamental Research, Mumbai, Maharashtra, India

    Anish Ghosh

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Pollicott, M. (2019). Exponential Mixing: Lectures from Mumbai. In: Dani, S., Ghosh, A. (eds) Geometric and Ergodic Aspects of Group Actions. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-15-0683-3_4

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