arXiv subjects

Artur O. Lopes

Publications and source records attributed to Artur O. Lopes.

At least 37 records · Page 2Linked to original sources

arXiv2022

Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in a recent paper by K. McGoff, S. Mukherjee and A. Nobel. Our approach makes use of non-additive thermodynamic formalism and large deviation properties instead of joinings.

math.ST↗

arXiv2021

Decision Theory and Large Deviations for Dynamical Hypotheses Tests: Neyman-Pearson, Min-Max and Bayesian Tests

We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size $n$ goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.

math.ST↗

arXiv2021

Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry

In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a Hölder Jacobian $J$ and the Ruelle (transfer) operator $\mathcal{L}_{\log J}.$ Two equilibrium probabilities $μ_1$ and $μ_2$, can interact via a discrete-time {\it Thermodynamic Operation} described by the action {\it of the dual of the Ruelle operator} $ \mathcal{L}_{\log J}^*$. We argue that the law $μ\to \mathcal{L}_{\log J}^*(μ)$, producing nonequilibrium, can be seen as a Thermodynamic Operation after showing that it's a manifestation of the Second Law of Thermodynamics. We also show that the change of relative entropy satisfies $$ D_{K L} (μ_1,μ_2) - D_{K L} (\mathcal{L}_{\log J}^*(μ_1),\mathcal{L}_{\log J}^*(μ_2))= 0.$$ Furthermore, we describe sufficient conditions on $J,μ_1$ for getting $h(\mathcal{L}_{\log J}^*(μ_1))\geq h(μ_1)$, where $h$ is entropy. Recalling a natural Riemannian metric in the Banach manifold of Hölder equilibrium probabilities we exhibit the second-order Taylor formula for an infinitesimal tangent change of KL divergence; a crucial estimate in Information Geometry. We introduce concepts like heat, work, volume, pressure, and internal energy, which play here the role of the analogous ones in Thermodynamics of gases. We briefly describe the MaxEnt method.

math.DS↗

arXiv2021

C*- Algebras and Thermodynamic Formalism

We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {\it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1051-1082 (2004). We show only the uniqueness of the β-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $μ$) and its relation with the eigen-probability $ν_β$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for the Gibbs state. There is no Phase transition for the KMS state.

math.DS↗

arXiv2021

Lyapunov exponents for Quantum Channels: an entropy formula and generic properties

We denote by $M_k$ the set of $k$ by $k$ matrices with complex entries. We consider quantum channels $ϕ_L$ of the form: given a measurable function $L:M_k\to M_k$ and a measure $μ$ on $M_k$ we define the linear operator $ϕ_L:M_k \to M_k$, by the law $ρ\,\to\,ϕ_L(ρ) = \int_{M_k} L(v) ρL(v)^\dagger \, \dm(v).$ On a previous work the authors show that for a fixed measure $μ$ it is generic on the function $L$ the $Φ$-Erg property (also irreducibility). Here we will show that the purification property is also generic on $L$ for a fixed $μ$. Given $L$ and $μ$ there are two related stochastic process: one takes values on the projective space $ P(\C^k)$ and the other on matrices in $M_k$. The $Φ$-Erg property and the purification condition are good hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically on $L$, if $\int |L(v)|^2 \log |L(v)|\, \ dμ(v)<\infty$, then the Lyapunov exponents $\infty > γ_1\geq γ_2\geq ...\geq γ_k\geq -\infty$ are well defined. On the previous work it was presented the concepts of entropy of a channel and of Gibbs channel; and also an example (associated to a stationary Markov chain) where this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the above mentioned example and we show that it is equal to $-\frac{1}{2} \,h$, where $h$ is the entropy of the associated Markov probability.

math.DS↗

arXiv2021

Entropy, pressure, ground states and calibrated sub-actions for linear dynamics

Denote by $X$ a Banach space and by $T : X \to X$ a bounded linear operator with non-trivial kernel satisfying suitable conditions. We consider the concepts of entropy - for $T$-invariant probability measures - and pressure for Hölder continuous potentials. We also prove the existence of ground states (the limit when temperature goes to zero) associated with such class of potentials when the Banach space $X$ is equipped with a Schauder basis. We produce an example concerning weighted shift operators defined on the Banach spaces $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$, $1 \leq p < +\infty$, where our results do apply. In addition, we prove the existence of calibrated sub-actions when the potential satisfies certain regularity conditions using properties of the so-called Mañé potential. We also exhibit examples of selection at zero temperature and explicit sub-actions in the class of Hölder continuous potentials.

math.DS↗

arXiv2021

Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

Denote $M_k$ the set of complex $k$ by $k$ matrices. We will analyze here quantum channels $ϕ_L$ of the following kind: given a measurable function $L:M_k\to M_k$ and the measure $μ$ on $M_k$ we define the linear operator $ϕ_L:M_k \to M_k$, via the expression $ρ\,\to\,ϕ_L(ρ) = \int_{M_k} L(v) ρ{L(v)}^\dagger \, \dm(v)$. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where $L$ was the identity. Under some mild assumptions on the quantum channel $ϕ_L$ we analyze the eigenvalue property for $ϕ_L$ and we define entropy for such channel. For a fixed $μ$ (the \textit{a priori} measure) and for a given a Hamiltonian $H: M_k \to M_k$ we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such $H$) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed $μ$ (with more than one point in the support) the set of $L$ such that it is $ϕ$-Erg (also irreducible) for $μ$ is a generic set. We describe a related process $X_n$, $n\in \mathbb{N}$, taking values on the projective space $ P(\C^k)$ and analyze the question of the existence of invariant probabilities. We also consider an associated process $ρ_n$, $n\in \mathbb{N}$, with values on $\mathcal{D}_k$ ($\mathcal{D}_k$ is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator which is fixed for $ϕ_L$.

math.DS↗

arXiv2021

Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

We show the existence of invariant ergodic $σ$-additive probability measures with full support on $X$ for a class of linear operators $L: X \to X$, where $L$ is a weighted shift operator and $X$ either is the Banach space $c_0(\mathbb{R})$ or $l^p(\mathbb{R})$ for $1\leq p<\infty$. In order to do so, we adapt ideas from Thermodynamic Formalism as follows. For a given bounded Hölder continuous potential $A:X \to \mathbb{R}$, we define a transfer operator $\mathcal{L}_A$ which acts on continuous functions on $X$ and prove that this operator satisfies a Ruelle-Perron-Frobenius theorem. That is, we show the existence of an eigenfunction for $\mathcal{L}_A$ which provides us with a normalized potential $\overline{A}$ and an action of the dual operator $\mathcal{L}_{\overline{A}}^*$ on the $1$-Wasserstein space of probabilities on $X$ with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of $\mathcal{L}_A$ requires an {\it a priori} probability on the kernel of $L$. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.

math.DS↗

arXiv2021

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

It is well known that in Information Theory and Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an {\it a priori} probability kernel $\hatν$ and a probability $π$ on the measurable space $X\times Y$ we consider an appropriate definition of entropy of $π$ relative to $\hatν$, which is based on previous works. Using this concept of entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence in the case $\hatν$ is identified with a probability $ν$ on $X$. This will be used to extend the meaning of specific information gain and dynamical entropy production to the model of thermodynamic formalism for symbolic dynamics over a compact alphabet (TFCA model). In this case, we show that the involution kernel is a natural tool for better understanding some important properties of entropy production.

math.DS↗

arXiv2021

Pentadiagonal Matrices and an Application to the Centered MA(1) Stationary Gaussian Process

In this work, we study the properties of a pentadiagonal symmetric matrix with perturbed corners. More specifically, we present explicit expressions for characterizing when this matrix is non-negative and positive definite in two special and important cases. We also give a closed expression for the determinant of such matrices. Previous works present the determinant in a recurrence form but not in an explicit one. As an application of these results, we also study the limiting cumulant generating function associated to the bivariate sequence of random vectors (n^{-1} (\sum_{k=1}^n X_k^2 , \sum_{k=2}^n X_k X_{k-1})_{n in N}, when (X_n)_{n in N} is the centered stationary moving average process of first order with Gaussian innovations. We exhibit the explicit expression of this limiting cumulant generating function. Finally, we present three examples illustrating the techniques studied here.

math.PR↗

arXiv2020

Explicit examples in Ergodic Optimization

Denote by $T$ the transformation $T(x)= 2 \,x $ (mod 1). Given a potential $A:S^1 \to \mathbb{R}$ we are interested in exhibiting in several examples the explicit expression for the calibrated subaction $V: S^1 \to \mathbb{R}$ for $A$. The action of the $1/2$ iterative procedure $\mathcal{G}$, acting on continuous functions $f: S^1 \to \mathbb{R}$, was analyzed in a companion paper. Given an initial condition $f_0$, the sequence, $\mathcal{G}^n(f_0)$ will converge to a subaction. The sharp numerical evidence obtained from this iteration allow us to guess explicit expressions for the subaction in several worked examples: among them for $A(x) = \sin^2 ( 2 πx)$ and $A(x) = \sin ( 2 πx)$. Here, among other things, we present piecewise analytical expressions for several calibrated subactions. The iterative procedure can also be applied to the estimation of the joint spectral radius of matrices. We also analyze the iteration of $\mathcal{G}$ when the subaction is not unique. Moreover, we briefly present the version of the $1/2$ iterative procedure for the estimation of the main eigenfunction of the Ruelle operator.

math.DS↗

arXiv2020

An iterative process for approximating subactions

We describe a procedure based on the iteration of an initial function by an appropriated operator, acting on continuous functions, in order to get a fixed point. This fixed point will be a calibrated subaction for the doubling map on the circle and a fixed Lipschitz potential. We study analytical and generic properties of this process and we provide some computational evaluations of subactions using a discretization of the circle. The fixed point is unique if the maximizing probability is unique. We proceed a careful analysis of the dynamics of this operator close by the fixed point in order to explain the difficulty in estimating its asymptotic behavior. We will show that the convergence rate can be in some moments like $1/2$ and sometimes arbitrarily close to $1$.

math.DS↗

arXiv2020

Thermodynamic Formalism for Haar systems in Noncommutative Integration: transverse functions and entropy of transverse measures

We consider here a class of groupoids obtained via an equivalence relation (the subgroupoids of pair groupoids). We generalize to Haar Systems in these groupoids some results related to entropy and pressure which are well known in Thermodynamic Formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the {\it a priori} probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Our results are on measurable category.

math.DS↗

arXiv2019

Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures. One of the main results of the article is to show that a probability is DLR-Gibbs (associated to a continuous translation invariant specification), if and only if, is an eigenprobability for the transpose of the Ruelle operator. Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present

math.DS↗

arXiv2019

The Ruelle operator for symmetric $β$-shifts

Consider $m \in \mathbb{N}$ and $β\in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $β$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)β^{-n}$ where the sequence $x = (x(n))_{n \in \mathbb{N}}$ take values in the alphabet $\mathcal{A}_m := \{0, \ldots, m\}$. The above expression is called the $β$-expansion of $a$ and it is not necessarily unique. We are interested in sequences $x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$ which are associated to all possible values $a$ which have a unique expansion. We denote the set of such $x$ (with some more technical restrictions) by $X_{m,β} \subset\mathcal{A}_m^\mathbb{N}$. The space $X_{m, β}$ is called the symmetric $β$-shift associated to the pair $(m, β)$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential $A:X_{m, β} \to\mathbb{R}$, we consider the Ruelle operator $\mathcal{L}_A$ and we show the existence of a positive eigenfunction $ψ_A$ and an eigenmeasure $ρ_A$ for some appropriated values of $m$ and $β$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $h(μ_{tA})_{t>1}$ converges, when $t \to\infty$, to the maximal value among the set of all possible values of entropy of all $A$-maximizing probabilities.

math.DS↗

arXiv2019

Gibbs States and Gibbsian Specifications on the space $\mathbb{R}^{\mathbb{N}}$

We are interested in the study of Gibbs and equilbrium probabilities on the lattice $\mathbb{R}^{\mathbb{N}}$. Consider the unilateral full-shift defined on the non-compact set $\mathbb{R}^{\mathbb{N}}$ and an $α$-Hölder continuous potential $A$ from $\mathbb{R}^{\mathbb{N}}$ into $\mathbb{R}$. From a suitable class of a priori probability measures $ν$ (over the Borelian sets of $\mathbb{R}$) we define the Ruelle operator associated to $A$ (using an adequate extension of this operator to the compact set $\overline{\mathbb{R}}^\mathbb{N}=(S^1)^\mathbb{N}$) and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to $A$. We are also able to show several of the well known classical properties of Thermodynamic Formalism for both of these probability measures. The above, can be seen as a generalization of the results obtained in the compact case for the XY-model. We also introduce an extension of the definition of entropy and show the existence of $A$-maximizing measures (via ground states for $A$); we show the existence of the zero temperature limit under some mild assumptions. Moreover, we prove the existence of an involution kernel for $A$ (this requires to consider the bilateral full-shift on $\mathbb{R}^{\mathbb{Z}}$). Finally, we build a Gibbsian specification for the Borelian sets on the set $\mathbb{R}^{\mathbb{N}}$ and we show that this family of probability measures satisfies a \emph{FKG}-inequality.

math.DS↗

arXiv2019

Haar systems, KMS states on von Neumann algebras and $C^*$-algebras on dynamically defined groupoids and Noncommutative Integration

We analyse certain Haar systems associated to groupoids obtained by certain natural equivalence relations of dynamical nature on sets like $\{1,2,...,d\}^\mathbb{Z}$, $\{1,2,...,d\}^\mathbb{N}$, $S^1\times S^1$, or $(S^1)^\mathbb{N}$, where $S^1$ is the unitary circle. We also describe properties of transverse functions, quasi-invariant probabilities and KMS states for some examples of von Neumann algebras (and also $C^*$-Algebras) associated to these groupoids. We relate some of these KMS states with Gibbs states of Thermodynamic Formalism. While presenting new results, we will also describe in detail several examples and basic results on the above topics. In other words it is also a survey paper. Some known results on non-commutative integration are presented, more precisely, the relation of transverse measures, cocycles and quasi-invariant probabilities. We describe the results in a language which is more familiar to people in Dynamical Systems. Our intention is to study Haar systems, quasi-invariant probabilities and von Neumann algebras as a topic on measure theory (intersected with ergodic theory) avoiding questions of algebraic nature (which, of course, are also extremely important).

math.DS↗

arXiv2018

Continuous groupoids on the symbolic space, quasi-invariant probabilities for Haar systems and the Haar-Ruelle operator

We consider groupoids on $\{1,2,..,d\}^\mathbb{N}$, cocycles and the counting measure as transverse function. We generalize results relating quasi-invariant probabilities with eigenprobabilities for the dual of the Ruelle operator. We assume a mild compatibility of the groupoid with the symbolic structure. We present a generalization of the Ruelle operator - the Haar-Ruelle operator - taking into account the Haar structure. We consider continuous and also Hölder cocycles. IFS with weights appears in our reasoning in the Hölder case.

math.DS↗