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L. Cioletti

Publications and source records attributed to L. Cioletti.

9 recordsLinked to original sources
arXiv2020

On the Dimension of the Space of Harmonic Functions on Transitive Shift Spaces

In this paper, we show a new relation between phase transition in one-dimensional Statistical Mechanics and the multiplicity of the dimension of the space of harmonic functions for an extension of the classical transfer operator. We accomplish this by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials. This is done by establishing finer properties of the associated conformal measures and thoroughly developing a method to obtain information on the maximal eigenspace of a suitably constructed family of Markov Processes. Our results are valid in the setting of finite and infinite alphabets. Several new applications are given to illustrate the theory. For example, we determine the support of a large class of equilibrium states associated with low regular potentials, including ones allowing phase transition. Additionally, we prove a version of the Functional Central Limit Theorem for equilibrium states. A remarkable aspect of this result is that it does not require the spectral gap property of the associated transfer operator. It is valid for long-range spins systems that might not be positively correlated and for non-local observables.

math.DS↗
arXiv2020

The Double Transpose of the Ruelle Operator

In this paper we study the double transpose of the $L^1(X,\mathscr{B}(X),ν)$-extensions of the Ruelle transfer operator $\mathscr{L}_{f}$ associated to a general real continuous potential $f\in C(X)$, where $X=E^{\mathbb{N}}$, the alphabet $E$ is any compact metric space and $ν$ is a maximal eigenmeasure. For this operator, denoted by $\mathbb{L}^{**}_{f}$, we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $ρ(\mathscr{L}_{f})$, the spectral radius of the Ruelle operator acting on $C(X)$. As an application, we obtain a sufficient condition ensuring that the natural extension of the Ruelle operator to $L^1(X,\mathscr{B}(X),ν)$ has an eigenfunction associated to $ρ(\mathscr{L}_{f})$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ belongs to the Hölder, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.

math.DS↗
arXiv2019

Thermodynamic Formalism for Topological Markov Chains on Borel Standard Spaces

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $X\equiv E^{\mathbb{N}}$, where $E$ is a general Borel standard space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on $X$ and obtain the existence of equilibrium states as additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states do not have a purely additive part. We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite dimensional diffusion.

math.DS↗
arXiv2018

Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems

We study existence and uniqueness of the fixed points solutions of a large class of non-linear variable discounted transfer operators associated to a sequential decision-making process. We establish regularity properties of these solutions, with respect to the immediate return and the variable discount. In addition, we apply our methods to reformulating and solving, in the setting of dynamic programming, some central variational problems on the theory of iterated function systems, Markov decision processes, discrete Aubry-Mather theory, Sinai-Ruelle-Bowen measures, fat solenoidal attractors, and ergodic optimization.

math.DS↗
arXiv2017

Limit Theorems in Mallows Distance for Processes with Gibssian Dependence

In this paper, we explore the connection between convergence in distribution and Mallows distance in the context of positively associated random variables. Our results extend some known invariance principles for sequences with FKG property. Applications for processes with Gibbssian dependence structures are included.

math.PR↗
arXiv2017

Thermodynamic Formalism for Iterated Function Systems with Weights

This paper introduces an intrinsic theory of Thermodynamic Formalism for Iterated Functions Systems with general positive continuous weights (IFSw).We study the spectral properties of the Transfer and Markov operators and one of our first results is the proof of the existence of at least one eigenprobability for the Markov operator associated to a positive eigenvalue. Sufficient conditions are provided for this eingenvalue to be the spectral radius of the transfer operator and we also prove in this general setting that positive eigenfunctions of the transfer operator are always associated to its spectral radius. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSw's with weights given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional. We also recover the classical formula relating the powers of the transfer operator and the topological pressure and establish its uniform convergence. In the last section we present some examples and show that the results obtained can be viewed as a generalization of several classical results in Thermodynamic Formalism for ordinary dynamical systems.

math.DS↗
arXiv2017

Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if $f$ does not satisfy Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N} \to \mathbb{R}$ of the form $$ f(x_1,x_2,\ldots) = x_1 + 2^{-γ} \, x_2 + 3^{-γ} \, x_3 + ...+n^{-γ} \, x_n + \ldots $$ with $γ>1$. For $3/2 < γ\leq 2$, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.

math.DS↗
arXiv2016

Bounds on the Quenched Pressure and Main Eigenvalue of the Ruelle Operator for Brownian Type Potentials

In this paper we consider a random potential derived from the Brownian motion. We obtain upper and lower bounds for the expected value of the main eigenvalue of the associated Ruelle operator and for its quenched topological pressure. We also exhibit an isomorphism between the space $C(Ω)$ endowed with its standard norm and a proper closed subspace of the Skorokhod space which is used to obtain a stochastic functional equation for the main eigenvalue and for its associated eigenfunction.

math.DS↗