Search arXiv⌕ Search

arXiv subjects

Daniel Coronel

Publications and source records attributed to Daniel Coronel.

11 recordsLinked to original sources
arXiv2026

Thermodynamic formalism for intermittent maps with multiple neutral fixed points and phase transitions

In this paper, we develop the thermodynamic formalism for intermittent full-branch interval maps with finitely many neutral fixed points and for a broad class of possibly discontinuous potentials. Under a hyperbolicity assumption on the potential, we prove the existence of a unique equilibrium state with positive entropy and exponential decay of correlations. Furthermore, we characterize the phase transitions for the one-parameter family $\beta\varphi$, with $\beta>0$, and prove that, at the critical parameter, multiple ergodic equilibrium states may coexist, at most one of which has positive entropy. Our approach relies on a detailed analysis of conformal measures.

math.DS↗
arXiv2025

Phase diagram for intermittent maps

We explore the phase diagram for potentials in the space of H\"older continuous functions of a given exponent and for the dynamical system generated by a Pomeau--Manneville, or intermittent, map. There is always a phase where the unique Gibbs state exhibits intermittent behavior. It is the only phase for a specific range of values of the H\"older exponent. For the remaining values of the H\"older exponent, a second phase with stationary behavior emerges. In this case, a co-dimension 1 submanifold separates the intermittent and stationary phases. It coincides with the set of potentials at which the pressure function fails to be real-analytic. We also describe the relationship between the phase transition locus, (persistent) phase transitions in temperature, and ground states.

math.DS↗
arXiv2025

Phase transitions in temperature for intermittent maps

This article characterizes phase transitions in temperaturefor intermittent maps within a specific space of \textsc{H\"older} continuous potentials, distinguished by their regularity and asymptotic behavior at zero. We also characterize the phase transitions in temperature that are robust within this space. Our results reveal a connection between phase transitions in temperature and ergodic optimization.

math.DS↗
arXiv2020

Model sets with Euclidean internal space

We give an almost dynamical characterization of inter-model sets with Euclidean internal space. This characterization is similar to previous results for general inter-model sets obtained independently by Baake, Lenz and Moody, and Aujogue. The new ingredients are two additional conditions on the rank of the Abelian group generated by the set of internal difference and a flow on a torus defined via the address map introduced by Lagarias that play the role of the maximal equicontinuous factor in the previous characterizations.

math.DS↗
arXiv2018

Sensitive dependence of geometric Gibbs states at positive temperature

We give the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states do not converge at positive temperature. These are the first examples of non-convergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel. We also show that this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states at positive temperature.

math.DS↗
arXiv2017

Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps

For quadratic-like maps, we show a phenomenon of sensitive dependence of geometric Gibbs states: There are analytic families of quadratic-like maps for which an arbitrarily small perturbation of the parameter can have a definite effect on the low-temperature geometric Gibbs states. Furthermore, this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states. We introduce a geometric version of the Peierls condition for contour models ensuring that the low-temperature geometric Gibbs states are concentrated near the critical orbit.

math.DS↗
arXiv2014

Sensitive dependence of Gibbs measures at low temperatures

The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

math-ph↗
arXiv2013

High-order phase transitions in the quadratic family

We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, near the phase transition these functions behave as $x \mapsto \exp (- 1 / x^2)$ near $x = 0$, before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

math.DS↗
arXiv2012

Low-temperature phase transitions in the quadratic family

We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

math.DS↗
arXiv2010

Tower systems for Linearly repetitive Delone sets

In this paper we study linearly repetitive Delone sets and prove, following the work of Bellissard, Benedetti and Gambaudo, that the hull of a linearly repetitive Delone set admits a properly nested sequence of box decompositions (tower system) with strictly positive and uniformly bounded (in size and norm) transition matrices. This generalizes a result of Durand for linearly recurrent symbolic systems. Furthermore, we apply this result to give a new proof of a classic estimation of Lagarias and Pleasants on the rate of convergence of patch-frequencies.

math.DS↗