Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

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.

Ledrappier–Young formulae for a family of nonlinear attractors

“…We will study the pushforward measure μ = m • −1 for an ergodic invariant quasi-Bernoulli measure m, noting that μ is supported on F. Apart from including the important class of Bernoulli measures, quasi-Bernoulli measures also include the well-known class of Gibbs measures for Hölder continuous potentials. Furthermore it was shown in [2] that this inclusion is strict. The Shannon-McMillan-Breiman theorem allows us to define the entropy of μ.…”

Dimension spectrum of infinite self-affine iterated function systems

“…Note that [17, Proposition 2.8] ensures quasimultiplicativity for all values of s only in the planar case because in this setting the singular value function can be written as in (3). In higher dimensions the quasimultiplicativity of the norm implies quasimultiplicativity of φ s for s ∈…”

“…Recently in [3], Bárány, Morris and Käenmäki studied more general properties of the semigroup generated by A which ensure almost multiplicativity of the norm.…”

Totally Ergodic Generalised Matrix Equilibrium States have the Bernoulli Property

“…(In other literatures this property is sometimes called local product structure: see for example [11]). It follows from the results of [10] that every ergodic generalised matrix equilibrium state satisfies the upper bound μ([ij]) ≤ Cμ([i])μ ([j]), but the lower bound does not hold in general (see for example [5]). If T 1 , .…”