Contributions to phase transition of intermittent skew-product and piecewise monotone dynamics on the circle - Silva, Afonso Fernandes da (2023)
It is known that any uniformly dusty or hyperbolic transitive dynamics do not it has phase transition with respect to the continuous Hölder potentials. When it comes to more general dynamics, it is still an open question to classify all the dynamics that They have transition with respect to a certain class of regular potentials. In dimension 1, according to Bomfim-Carneiro [BC21], every local C1+α-diffeomorphism in the transitive circle that is neither expander nor invertible has a single thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function R ∋ t 7→ Ptop(f, −tlog |Df|) is analytic except at a point t0 ∈ (0, 1). They also proved spectral phase transition, i.e., the transfer operator Lf,−tlog |Df| Acting in the space of continuous Hölder functions, there is a spectral gap for all t < t0 and There is no spectral gap for T ≥ T0. Our goal is to prove similar results for two special classes of dynamics: partially codimensional 1 endomorphisms hyperbolic and monotonous dynamics by parts in the circle transitive. For high-dimensional endomorphisms, we proved that the thermodynamic and spectral phase transition results imply multifractal analysis for the Lyapunov spectrum. In particular We exhibit a class of partially hyperbolic endomorphisms that admit transition of thermodynamic and spectral phase with respect to the geometric potential in the central direction, and we describe multifractal analysis of the central Lyapunov exponents. For monotonous dynamics by parts in the circle, we prove that the set of continuous Hölder potentials that do not have thermodynamic and spectral phase transition is dense in the uniform topology and the set of continuous Hölder potentials that have phase transition is not dense in the uniform topology. We also get a transitional characterization in terms of the transfer operator and the convexity type of the topological pressure function. In In particular, we describe the behavior of the topological pressure function and the associated transfer.