2021/06/15 by Thiago Bomfim, Bomfim, Thiago, Victor Carneiro +1 · 1 citation
Mathematics · Physics and Astronomy · #37C30 #37C40 #37D35 #37E10 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2106.08436
openalex publication_date 2021/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that all uniformly expanding dynamics have no phase transition with respect to Hölder continuous potentials. In this paper we show that given a local diffeomorphism f on the circle, that is neither a uniformly expanding dynamics nor invertible, the topological pressure function ℝ \ni t ↦ Ptop(f , -tlog |Df|) is not analytical. In other words, f has a thermodynamic phase transition with respect to geometric potential. Assuming that f is transitive and that Df is Hölder continuous, we show that there exists t0 ∈ (0 , 1] such that the transfer operator Lf, -tlog|Df|, acting on the space of Hölder continuous functions, has the spectral gap property for all t < t0 and has not the spectral gap property for all t ≥ t0. Similar results are also obtained when the transfer operator acts on the space of bounded variations functions and smooth functions. In particular, we show that in the transitive case f has a unique thermodynamic phase transition and it occurs in t0. In addition, if the loss of expansion of the dynamics occurs because of an indifferent fixed point or the dynamics admits an absolutely continuous invariant probability with positive Lyapunov exponent then t0 = 1.