Thermodynamic formalism of \text{GL}_2(\mathbb{R}) -cocycles with canonical holonomies

Abstract

We study the norm potentials of Hölder continuous GL 2 (R)-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant GL 2 (R)-cocycles as well as fiber-bunched GL 2 (R)-cocycles. We show that the norm potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose norm potentials have more than one equilibrium states.

Totally Ergodic Generalised Matrix Equilibrium States have the Bernoulli Property

“…, U r j j } by U j (x) := U i j if x ∈ Z ∩ X i and U j (x) := U 1 j otherwise, it is obvious that U j is a measurable function. The equation ( 17) precisely asserts the truth of ( 14) and combining this with (12), (13) and the definition of Z yields (15). In particular ( 14) and ( 15) hold for all x ∈ Z ∩ (X 1 ∪ • • • ∪ X r j ).…”

“…However, the full range of possible behaviours outside this class of repellers is far from being completely understood even in the self-affine case. Beyond the self-affine class we anticipate that it might not be difficult to extend our methods and results to the case of typical repellers which satisfy a fibre-bunching condition on the derivative cocycle (x, n) → D x f n as in [15,21,47], particularly if a strong additional assumption is used such as the "pinching and twisting" conditions introduced by Bonatti and Viana [11]. The removal of the fibre-bunching condition may be a more substantial obstacle to further development of these ideas.…”

TheK-property for subadditive equilibrium states

“…For the first statement, it was shown by Bochi and Garibaldi [BG19] that A is spannable. The second author and Butler [BP19] then showed that the norm potentials of spannable cocycles are quasi-multiplicative. The second statement is established by the second author in [Par20].…”