Searcharxiv⌕ Search

arXiv subjects

Renaud Leplaideur

Publications and source records attributed to Renaud Leplaideur.

At least 19 recordsLinked to original sources
arXiv2025

Freezing phase transition for the Thue-Morse subshift

On the full shift on two symbols, we consider the potential defined by $V(x) = \frac{1}{n}$ where $n$ denotes the longest common prefix between the infinite word $x$ and an element of the subshift associated to the Thue-Morse substitution. Given a non negative real number $β$, the pressure function is $P(β):=\sup\left\{h_μ+β\int V\,dμ\right\},$ where the supremum is taken over all shift invariant probabilities $μ$ on the full shift and $h_μ$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $β$ large enough, the pressure $P(\be)$ is equal to zero. Similar results were previously published by Bruin and Leplaideur in \cite{BL2}, \cite{Bruin-Leplaid-13} but their proofs contained significant gaps and required substantial clarification.

math.DS↗
arXiv2025

Limits of equilibrium states for coupled weakly interacting systems. Application to the measure of maximal entropy

We study metastability for symbolic dynamic. We prove that for a global system given by two independent sub-systems linked by a hole, and for a Lipschitz continuous potential, the global equilibrium state converges, as the hole shrinks, to a convex combination of the two independent equilibria in each component. Two kinds of convergence occur, depending on the assumptions on how long an orbit has to stay in each well. As a by-product, we show that this can be applied to a geometrical system inspired from [11] and for the measure of maximal entropy.

math.DS↗
arXiv2025

On the speed of convergence of the pressure function at zero temperature

We prove here that the pressure function cannot converge to the limit entropy at zero temperature faster than some exponential rate. Furthermore, we characterize this limit rate via an expression involving the Peierls barriers between the irreducible components of the Aubry set. This extends and completes results from [8] and [7]. In the first one, an exact exponential speed of convergence was proved, under the assumption that the Aubry set is a subshift of finite type. In the later one, a rate was given but without interpretation in term of Thermodynamical quantities of the system.

math.DS↗
arXiv2024

On the selection of subaction and measure for perturbed potentials

We prove that when the Aubry set for a Lipschitz continuous potential is a subshift of finite type, then the pressure function converges exponentially fast to its asymptote as the temperature goes to 0. The speed of convergence turns out to be the unique eigenvalue for the matrix whose entries are the costs between the different irreducible pieces of the Aubry set. For a special case of Walter potential we show that pertubation of that potential that go faster to zero than the pressure do not change the selection, nor for the subaction, neither for the limit measure a zero temperature.

math.DS↗
arXiv2023

An Estimation of Phase Transition

In [8], H. Bruin and R. Leplaideur studied a class of potentials such that the pressure function exhibit a phase transition at a parameter \b{eta}c > 0. This paper will prove that the transition in pressure function cannot appear within the interval ]0, 2].

math.DS↗
arXiv2023

The Jacaranda tree is strongly aperiodic and has zero entropy

We prove that the Jacaranda tree obtained as a fixed point for a substreetution in previous work of the authors is strongly aperiodic and that the number of patches increases linearly with respect to the size of the patch. As a consequence we get that the tree has zero entropy.

math.DS↗
arXiv2021

Nonlinear thermodynamical formalism

We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending investigation of the quadratic case by Leplaideur and Watbled. Under suitable conditions, we prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures. In this non-linear thermodynamical formalism, which can, e.g., model mean-field approximation of large systems, several kind of phase transitions appear, some of which cannot happen in the linear case. We use our correspondence between non-linear and linear equilibrium measures to further the understanding of phase transitions, { both in previously known cases (Curie-Weiss and Potts models) and in} new examples (metastable phase transition). Finally, we apply some of the ideas introduced to the classical thermodynamical formalism, proving that freezing phase transitions can occur over \emph{any} zero-entropy invariant compact subset of the phase space.

math.DS↗
arXiv2019

Uniqueness of the measure of maximal entropy for singular hyperbolic flows in dimension 3 and more results on equilibrium states

We prove that any 3-dimensional singular hyperbolic attractor admits for any Hölder continuous potential $V$ at most one equilibrium state for $V$ among regular measures. We give a condition on $V$ which ensures that no singularity can be an equilibrium state. Thus, for these $V$'s, there exists a unique equilibrium state and it is a regular measure. Applying this for $V\equiv 0$, we show that any 3-dimensional singular hyperbolic attractor admits a unique measure of maximal entropy.

math.DS↗
arXiv2019

Thermodynamic formalism and Substitutions

This paper studies properties of a Renormalization Operator for potentials in symbolic dynamics. These operators first appeared in \cite{BLL} and the link with substitutions was done in \cite{BL1}. Their fixed points are natural candidates to have pathologic behavior such as phase transitions. If $R$ is such an operator, we study the convergence of $R^{n}(φ)$ to the non-nul fixed point. We define the family of marked substitutions, which contains the Thue-Morse substitution, and show that the associated renormalization operators on potentials admits a unique non-nul continuous fixed point. Then, we show that $R^{n}(φ)$ converges to the fixed point as soon as $φ$ has the right germ close to $\mathbb K$.

math.DS↗
arXiv2015

Kupka-Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a certain degree of freedom for the choice of the second one. For a non-countable set of choices, this leaf is not periodic and the diffeomorphism is Kupka-Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. As a consequence of our construction, the map is Hölder conjugated to a subshift of finite type, thus every Hölder potential admits a unique associated equilibrium state.

math.DS↗
arXiv2013

Renormalization, Freezing Phase Transitions and Fibonacci Quasicrystals

We examine the renormalization operator determined by the Fibonacci substitution. We exhibit a fixed point and determine its stable leaf (under iteration of the operator). Then, we study the thermodynamic formalism for po- tentials in this stable leaf, and prove they have a freezing phase transition, with ground state supported on the attracting quasi-crystal associated to the Fibonacci substitution

math.DS↗
arXiv2013

Srb Measures For Almost Axiom A Diffeomorphisms

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable and unstable leaves are "long enough", then f admits a probability SRB measure.

math.DS↗
arXiv2012

Thermodynamic formalism for Lorenz maps

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any Hölder continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.

math.DS↗
arXiv2012

Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator $\CR$ acting on the space of potentials are such that the pressure function $γ\mapsto \CP(-γV)$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of $\CR$, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.

math.DS↗
arXiv2012

Flatness is a Criterion for Selection of Maximizing Measures

For a full shift with Np+1 symbols and for a non-positive potential, locally proportional to the distance to one of N disjoint full shifts with p symbols, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest. In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes. As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction.

math.DS↗
arXiv2010

Selection of measures for a potential with two maxima at the zero temperature limit

For the subshift of finite type $§=\{0,1,2\}^{\N}$ we study the convergence at temperature zero of the Gibbs measure associated to a non-locally constant Hölder potential which admits only two maximizing measures. These measures are Dirac measures at two different fixed points. The potential is flattest at one of these two fixed points. The question we are interested is: which of these probabilities the invariant Gibbs state will select when temperature goes to zero? We prove that on the one hand the Gibbs measure converges, and at the other hand it does not necessarily converge to the measures where the potential is the flattest. We consider a family of potentials of the above form; for some of them there is the selection of a convex combination of the two Dirac measures, and for others there is a selection of the Dirac measure associated to the flattest point. In the first case this is contrary to what was expected if we consider the analogous problem in Aubry-Mather theory by N. Anantharaman, R. Iturriaga, P. Padilla and H. Sanchez-Morgado. The invariant probability is obtained by the junction of the eigen-function and the eigen-probability. A curious phenomena that happens in our examples is that the eigen-measure and the eigen-function have opposite behavior. When $β\to \infty$, the eigen-measure became exponential bigger around $0^\infty$, when compared to points around $1^\infty$. For the eigen-function the opposite happens.

math.DS↗
arXiv2009

Central limit theorem for dimension of Gibbs measures for skew expanding maps

We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius $\eps$ have a measure smaller than $\eps^δ$ and half of them have a measure larger than $\eps^δ$, where $δ$ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of the fluctuations of some Birkhoff sums. Then we use general results from probability theory as the weak invariance principle and random change of time to get our main theorem. Our method also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non uniformly expanding maps. These generalizations are presented at the end of the paper.

math.DS↗