Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles
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DOMINATION, ALMOST ADDITIVITY, AND THERMODYNAMIC FORMALISM FOR PLANAR MATRIX COCYCLES
BY
´zs Ba ´ ra ´ny Bala MTA-BME Stochastics Research Group, Department of Stochastics Budapest University of Technology and Economics P.O. Box 91, 1521 Budapest, Hungary e-mail: [email protected]
AND
¨enma ¨ki Antti Ka Department of Physics and Mathematics, University of Eastern Finland P.O. Box 111, FI-80101 Joensuu, Finland e-mail: [email protected]
AND
Ian D. Morris School of Mathematical Sciences, Queen Mary, University of London Mile End Road, London E1 4NS, United Kingdom e-mail: [email protected]
Received September 25 2018 and in revised form August 24, 2019
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´ ANY, ´ ¨ ¨ B. BAR A. KAENM AKI AND I. D. MORRIS
Isr. J. Math.
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.
1. Introduction For the purposes of this article a linear cocycle over a dynamical system T : X → X will be a skew-product F : X × Rd → X × Rd ,
(x, p) → (T x, A(x)p),
where A : X → GLd (R) is continuous and X is a compact metric space. Writing AnT (x) = A(T n−1 x) · · · A(x), we thus have F n (x, p) = (T n x, AnT (x)p) for all n ∈ N and (1.1)
n n ATm+n (x) = Am T (T x)AT (x)
for all m, n ∈ N. In numerous contexts it has been found useful to consider cocycles in which all of the matrices A(x) are positive: we note for example such diverse articles as [18, 19, 22, 30]. Under this assumption the cocycle satisfies the inequality n n log Am+n (x) − log Am T (T x) − log AT (x) C T for some constant C > 0 depending only on A. This has led some authors to extend results for positive linear cocycles by considering, instead of a linear
Vol. TBD, 2020
DOMINATION AND THERMODYNAMIC FORMALISM
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cocycle, a sequence of continuous functions fn : X → R satisfying the inequality |fn+m (x) − fm (T n x) − fn (x)| C for all x ∈ X and n, m 1. Such sequences of functions are referred to in the
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