Abstract
We show that every totally ergodic generalised matrix equilibrium state is psi -mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.
Highlights
Background and MotivationGiven a dynamical system f : M → M defined on a manifold M it is a matter of fundamental interest to be able to describe the behaviour of typical trajectories
There are situations in which this is insufficient: for example, a dynamical system may admit a repelling invariant set such that Lebesgue almost every point in an open neighbourhood of the invariant set eventually leaves that open set never to return; but it may still be of interest to understand which behaviours are typical among those points whose trajectories remain on the repelling set at all future times
The class of measures which we investigate in this article, which we call generalised matrix equilibrium states, are defined on abstract symbolic spaces and can be related to self-affine sets via a coding procedure which is described later
Summary
Given a dynamical system f : M → M defined on a manifold M it is a matter of fundamental interest to be able to describe the behaviour of typical trajectories. This makes the classical thermodynamic formalism of Bowen, Ruelle and Sinai, which applies to Hölder continuous real-valued potentials, applicable to the problem For this reason Conjecture 1 has for some time been substantially understood in the special case of repelling sets of conformal expanding maps in which all Lyapunov exponents of a given invariant measure are guaranteed to be equal. In particular the natural extension of every totally ergodic generalised matrix equilibrium state is measurably isomorphic to a Bernoulli process This completely resolves that part of Conjecture 1 which is concerned with mixing and the Bernoulli property in the special case where K is a self-affine set which is already known to support an invariant measure whose dimension is equal to a theoretical maximum value defined by Falconer [20]. The removal of the fibre-bunching condition may be a more substantial obstacle to further development of these ideas
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