From thermodynamic and spectral phase transitions to multifractal analysis

Thiago Bomfim; Victor Carneiro; Afonso Fernandes

Home/Archive/arXiv

arXiv · 2209.05590

From thermodynamic and spectral phase transitions to multifractal analysis

Thiago Bomfim·Victor Carneiro·Afonso Fernandes

View original source ↗Read source document ↗Export metadata ↓

Abstract

It is known that all uniformly expanding or hyperbolic dynamics have no phase transition with respect to H\"older continuous potentials. In \cite{BC21}, is proved that for all transitive $C^{1+\alpha}-$local diffeomorphism $f$ on the circle, that is neither a uniformly expanding map nor invertible, has a unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also it is proved spectral phase transitions, in other words, the transfer operator $\mathcal{L}_{f,-t\log|Df|}$ acting on the space of H\"older continuous functions, has the spectral gap property for all $t<t_0$ and does not have the spectral gap property for all $t\geq t_0$. Our goal is to prove that the results of thermodynamical and spectral phase transitions imply a multifractal analysis for the Lyapunov spectrum. In particular, we exhibit a class of partially hyperbolic endomorphisms that admit thermodynamical and spectral phase transitions with respect to the geometric potential, and we describe the multifractal analysis of your central Lyapunov spectrum.

Explore related subjects

math.DS

Keep this discovery

BibTeX RIS

Thiago Bomfim, Victor Carneiro, Afonso Fernandes. 2022-09-12. From thermodynamic and spectral phase transitions to multifractal analysis. https://arxiv.org/abs/2209.05590

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗

arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗