2023/08/31 by Tom Rush, Rush, Tom · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2308.16694
openalex publication_date 2023/08/31 · openalex created_date 2023/09/02 · openalex updated_date 2026/07/28
Let (ΣT,σ) be a subshift of finite type with primitive adjacency matrix T, ψ:ΣT → ℝ a Hölder continuous potential, and A:ΣT → GLd(ℝ) a 1-typical, one-step cocycle. For t ∈ ℝ consider the sequences of potentials Φt=(φt,n)n ∈ ℕ defined by φt,n(x):=Sn ψ(x) + tlog ‖An(x)‖, ∀ n ∈ ℕ. Using the family of transfer operators defined in this setting by Park and Piraino, for all t<0 sufficiently close to 0 we prove the existence of Gibbs-type measures for the superadditive sequences of potentials Φt. This extends the results of the well-understood subadditive case where t ≥ 0. Prior to this, Gibbs-type measures were only known to exist for t<0 in the conformal, the reducible, the positive, or the dominated, planar settings, in which case they are Gibbs measures in the classical sense. We further prove that the topological pressure function t ↦ Ptop(Φt,σ) is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.