arXiv subjects
Publications and source records attributed to Irene Inoquio-Renteria.
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.
In this work, we deal with Delone sets and their rectifiability under different classes of regularity. By pursuing techniques developed by Rivi\`ere and Ye, and Aliste-Prieto, Coronel and Gambaudo, we give sufficient conditions for a specific Delone set to be equivalent to the standard lattice by bijections having regularity in between bi-Lipschitz and bi-H\"older-homogeneous. From this criterion, we extend a result of McMullen by showing that, for any dimension $d\geq 1$, there exists a threshold of moduli of continuity $\mathcal{M}_d$, including the class of the H\"{o}lder ones, such that for every $\omega\in\mathcal{M}_d$, any two Delone sets in $\mathbb{R}^d$ within a certain class cannot be distinguished under bi-$\omega$-equivalence. This class accounts for the decreasing rate of the density deviation of a Delone set, with respect to a limit density. We also extend a result due to Aliste, Coronel, and Gambaudo, which establishes that every linearly repetitive Delone set in $\mathbb{R}^d$ is rectifiable, by extending it to a broader class of repetitive behaviors. Moreover, we show that for the modulus of continuity $\omega(t)=t(\log(1/t))^{1/d}$, every $\omega$-repetitive Delone set in this class is equivalent to the standard lattice by a bi-$\omega$-homogeneous map. Finally, we address a continuous problem related to the previous ones about finding solutions to the prescribed volume form equation in intermediate regularity, thereby extending the results of Rivi\`ere and Ye. Some interesting research directions are highlighted.
We provide a version of a thermodynamic formalism of entire transcendental maps that exhibit Baker domains, denoted as $f_{\ell, c}: \mathbb C\to \mathbb C$ and defined by $f_{\ell, c}(z)= c-(\ell-1)\log c+ \ell z- e^z$, where $\ell \in \mathbb N$, with $\ell \geq 2 $ and $c$ belongs to the disk $ D(\ell, 1)$ in the complex plane. We show in particular the existence and uniqueness of conformal measures and that the Hausdorff dimension is the unique zero of the pressure function $t\to P(t)$, for $t>1,$ where $J_r(f)$ is the radial Julia set.
In the context of expanding maps of the circle with an indifferent fixed point, understanding the joint behavior of dynamics and pairs of moduli of continuity $ (\omega, \Omega) $ may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus $ \Omega $ (precisely $ \lim_{x \to 0^+} \sup_{\mathsf d} \Omega\big({\mathsf d} x \big) / \Omega(\mathsf d) = 0 $) as a sufficient condition for the system to exhibit exponential decay of correlations with respect to the unique equilibrium state associated with a potential having $ \omega $ as modulus of continuity. This result is derived from obtaining the spectral gap property for the transfer operator acting on the space of observables with $ \Omega $ as modulus of continuity, a property that, as is well known, also ensures the Central Limit Theorem. Examples of application of our results include the Manneville-Pomeau family
In this note, we establish an original result for the thermodynamic formalism in the context of expanding circle transformations with an indifferent fixed point. For an observable whose continuity modulus is linked to the dynamics near such a fixed point, by identifying an appropriate linear space to evaluate the action of the transfer operator, we show that there is a strictly positive eigenfunction associated with the maximal eigenvalue given as the exponential of the topological pressure. Taking into account also the corresponding eigenmeasure, the invariant probability thus obtained is proved to be the unique Gibbs-equilibrium state of the system.
In ergodic optimization theory, the existence of sub-actions is an important tool in the study of the so-called optimizing measures. For transformations with regularly varying property, we highlight a class of moduli of continuity which is not compatible with the existence of continuous sub-actions. Our result relies fundamentally on the local behavior of the dynamics near a fixed point and applies to interval maps that are expanding outside an indifferent fixed point, including Manneville-Pomeau and Farey maps.
We identify a class of hyperbolic transcendental entire maps and we prove that some of its elements generate a class of potentials for which exhibit a conformal and invariant probability Gibbs measure. The methods and techniques from the thermodynamic formalism can be extended to this class of potentials. To complement this study we highlight that the dynamics of such a map on some subset of the Julia set is conjugated to the shift map over a code space with countable alphabet and the euclidean metric on the complex plane induces a metric on the symbolic space which is not compatible with the shift standard metric. From this fact, we provide a general description of the thermodynamic formalism from symbolic dynamic outlook, by studying the shift map acting on a non-compact and invariant subset of the full shift space with a countably infinite alphabet and a class of weakly H\"older continuous potentials, to prove the existence of a conformal and absolutely continuous invariant probability measure.
For a rational map $f$ and a Hölder continuous potential $ϕ$ satisfying $$ \supϕ< P(f,ϕ)$$ the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. For a given rational map $f$, in this paper we characterize those Hölder continuous potentials $ϕ$ for which this property is satisfied for some iterate of $f$.
Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a Hölder continuous potential $ϕ: J(f)\rightarrow \R$ and the pressure $P(f,ϕ). In the case where $\sup_{J(f)}ϕ<P(f,phi)$, the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials $ϕ$ for which this property is satisfied for some iterate of $f$, in terms of the expanding properties of the corresponding equilibrium states. A direct consequence of this result is that for a nonuniformly hyperbolic rational map every Hölder continuous potential has a unique equilibrium state and that this measure is exponentially mixing.