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Lyapunov Functions and Cone Families

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Abstract

We describe systematically the relation between Lyapunov functions and nonvanishing Lyapunov exponents, both for maps and flows. This includes a brief survey of the existing results in the area. In particular, we consider separately the cases of nonpositive and arbitrary Lyapunov functions, thus yielding optimal criteria for negativity and positivity of the Lyapunov exponents of linear cocycles over measure-preserving transformations. Moreover, we describe converse results of these criteria with the explicit construction of eventually strict Lyapunov functions for any map or flow with nonzero Lyapunov exponents. We also construct examples showing that in general the existence of an eventually strict invariant cone family does not imply the existence of an eventually strict Lyapunov function.

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Notes

  1. Kac’s lemma says that if f:X→X is a measurable transformation preserving a probability measure μ in X and E⊂X is a measurable set, then the function \(\bar{n}\) in (6) is μ-integrable and \(\int_{E} \bar{n} \, d \mu=\mu (\bigcup_{n \geq 0} f^{-n} E)\) (see for example [16]).

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Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, 1049-001, Lisbon, Portugal

    Luis Barreira, Davor Dragičević & Claudia Valls

Authors
  1. Luis Barreira
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  2. Davor Dragičević
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  3. Claudia Valls
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Correspondence to Luis Barreira.

Additional information

L.B. and C.V. are supported by the FCT grant PTDC/MAT/117106/2010 and by FCT through CAMGSD, Lisbon. D.D. is supported by the FCT grant SFRH/BD/78247/2011.

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Barreira, L., Dragičević, D. & Valls, C. Lyapunov Functions and Cone Families. J Stat Phys 148, 137–163 (2012). https://doi.org/10.1007/s10955-012-0524-8

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