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Domination, almost additivity and thermodynamical formalism for planar matrix cocycles

Bárány, Balázs, Käenmäki, Antti and Morris, Ian D. (2019) Domination, almost additivity and thermodynamical formalism for planar matrix cocycles Israel Journal of Mathematics.

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Abstract

In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Bárány, Balázs
Käenmäki, Antti
Morris, Ian D.i.morris@surrey.ac.uk
Date : 2019
Funders : Leverhulme Trust, European Research Council (ERC)
Copyright Disclaimer : © The Hebrew University of Jerusalem 2019
Uncontrolled Keywords : Products of matrices; Dominated splitting; Thermodynamic formalism; Almost additivity; Gibbs states
Related URLs :
Depositing User : Clive Harris
Date Deposited : 18 Sep 2019 09:56
Last Modified : 18 Sep 2019 09:56
URI: http://epubs.surrey.ac.uk/id/eprint/852661

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