arXiv subjects

Artur O. Lopes

Publications and source records attributed to Artur O. Lopes.

At least 73 records · Page 4Linked to original sources

arXiv2013

Selection of measure and a Large Deviation Principle for the general XY model

We consider $(M,d)$ a connected and compact manifold and we denote by $X$ the Bernoulli space $M^{\mathbb{N}}$. The shift acting on $X$ is denoted by $σ$. We analyze the general XY model, as presented in a recent paper by A. T. Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza. Denote the Gibbs measure by $μ_{c}:=h_{c}ν_{c}$, where $h_{c}$ is the eigenfunction, and, $ν_{c}$ is the eigenmeasure of the Ruelle operator associated to $cf$. We are going to prove that any measure selected by $μ_{c}$, as $c\to +\infty$, is a maximizing measure for $f$. We also show, when the maximizing probability measure is unique, that it is true a Large Deviation Principle, with the deviation function $R_{+}^{\infty}=\sum_{j=0}^\infty R_{+} (σ^f)$, where $R_{+}:= β(f) + V\circσ- V - f$, and, $V$ is any calibrated subaction.

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arXiv2013

A thermodynamic formalism for continuous time Markov chains with values on the Bernoulli Space: entropy, pressure and large deviations

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice 1,...,d}^N (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator $L={\mc L}_A -I$, where \mc L_A is a discrete time Ruelle operator (transfer operator), and A:{1,...,d}^N \to R is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the\emph{a priori}probability. Given a Lipschitz interaction V:\{1,...,d\}^{\bb N}\to \mathbb{R}, we are interested in Gibbs (equilibrium) state for such $V$. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative, and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer.

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arXiv2013

A Ruelle Operator for continuous time Markov Chains

We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $t\geq 0$, taking values on S. We denote by $Ω$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($Ω, {\cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${\cal L}^\infty (P)$. From the probability $P$ we define a Ruelle operator ${\cal L}^t, t\geq 0$, acting on functions $f:Ω\to \mathbb{R}$ of ${\cal L}^\infty (P)$. Given $V:Ω\to \mathbb{R}$, such that is constant in sets of the form $\{X_0=c\}$, we define a modified Ruelle operator $\tilde{\cal L}_V^t, t\geq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ν_V$ on $Ω$ associated to $\tilde{\cal L}^t_V, t\geq 0$. We also show the following property for the probability $ν_V$: for any integrable $g\in {\cal L}^\infty (P)$ and any real and positive $t$ $$ \int e^{-\int_0^t (V \circ Θ_s)(.) ds} [ (\tilde{\cal L}^t_V (g)) \circ θ_t ] d ν_V = \int g d ν_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).

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arXiv2012

Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics

We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume $\log g$ is well defined for this extension. It is known in Complex Dynamics that under the above hypothesis, for the given potential $β\,\log g$, where $β$ is a real constant, there exists a real analytic eigenfunction $ϕ_β$ defined on $[0,1]$ (with a complex analytic extension) for the Ruelle operator of $β\,\log g$. Under some assumptions we show that $\frac{1}β\, \log ϕ_β$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when $\log g(x)=-\log f'(x)$. In that case we relate the involution kernel to the so called scaling function.

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arXiv2012

Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $σ$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $Π$ the set of all Borel probabilities $π$ on $X\times X$, such that, both its $x$ and $y$ marginal are $σ$-invariant probabilities. We are interested in the optimal plan $π$ which minimizes $\int c d π$ among the probabilities on $Π$. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on $c$. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs $c$ the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan $π$ by convex combinations of plans such that the support projects in periodic orbits.

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arXiv2011

Wigner measures and the semi-classical limit to the Aubry-Mather measure

In this paper we investigate the asymptotic behavior of the semi-classical limit of Wigner measures defined on the tangent bundle of the one-dimensional torus. In particular we show the convergence of Wigner measures to the Mather measure on the tangent bundle, for energy levels above the minimum of the effective Hamiltonian. The Wigner measures $μ_h$ we consider are associated to $ψ_h,$ a distinguished critical solution of the Evans' quantum action given by $ψ_h=a_h\,e^{i\frac{u_h}h}$, with $a_h(x)=e^{\frac{v^*_h(x)-v_h(x)}{2h}}$, $u_h(x)=P\cdot x+\frac{v^*_h(x)+v_h(x)}{2},$ and $v_h,v^*_h$ satisfying the equations -\frac{h\, Δv_h}{2}+ 1/2 \, | P + D v_h \,|^2 + V &= \bar{H}_h(P), \frac{h\, Δv_h^*}{2}+ 1/2 \, | P + D v_h^* \,|^2 + V &= \bar{H}_h(P), where the constant $\bar{H}_h(P)$ is the $h$ effective potential and $x$ is on the torus. L.\ C.\ Evans considered limit measures $|ψ_h|^2$ in $\mathbb{T}^n$, when $h\to 0$, for any $n\geq 1$. We consider the limit measures on the phase space $\mathbb{T}^n\times\mathbb{R}^n$, for $n=1$, and, in addition, we obtain rigorous asymptotic expansions for the functions $v_h$, and $v^*_h$, when $h\to 0$.

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arXiv2011

Piecewise Analytic Subactions for Analytic Dynamics

We consider a piecewise analytic expanding map f: [0,1]-> [0,1] of degree d which preserves orientation, and an analytic positive potential g: [0,1] -> R. We address the analysis of the following problem: for a given analytic potential beta log g, where beta is a real constant, it is well known that there exists a real analytic (with a complex analytic extension to a small complex neighborhood of [0,1]) eigenfunction phi_beta for the Ruelle operator. One can ask: what happen with the function phi_beta, when beta goes to infinity. The domain of analyticity can change with beta. The correct question should be: is 1/ beta log phi_beta analytic in the limit, when beta goes to infinity ? Under a uniqueness assumption, this limit, when beta goes to infinity, is in fact a calibrated subaction V (see bellow definition). We show here that under certain conditions and for a certain class of generic potentials this continuous function is piecewise analytic (but not analytic). In a few examples one can get that the subaction is analytic (we need at least to assume that the maximizing probability has support in a unique fixed point).

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arXiv2010

The effective potential and transshipment in thermodynamic formalism at temperature zero

Denote the points in {1,2,..,r}^{Z}= {1,2,..,r}^{N} x {1,2,..,r}^{N} by ({y}^*, {x}). Given a Lipschitz continuous observable A: {1,2,..,r}^{Z} \to {R} , we define the map {G}^+: {H}\to {H} by {G}^+(ϕ)({y}^*) = \sup_{μ\in {M}_σ} [\int_{\{1,2,..,r\}^{N}} ( A({y}^*, {x}) + ϕ({x})) dμ({x}) + h_μ(σ) ], where: σis the left shift map acting on {1,2,..,r}^{N}; {M}_σdenotes the set of σ-invariant Borel probabilities; h_μ(σ) indicates the Kolmogorov-Sinai entropy; {H } is the Banach space of Lipschitz real-valued functions on {1,2,..,r}^{N}. We show there exist a unique ϕ^+ \in {H } and a unique λ^+\in {R} such that {G}^+ (ϕ^+) = ϕ^+ + λ^+. We say that ϕ^+ is the effective potential associated to A. This also defines a family of $σ$-invariant Borel probabilities μ_{{y}^*} on {1,2,..,r}^{N}, indexed by the points {y}^* \in {1,2,..,r}^{N}. Finally, for A fixed and for variable positive real values β, we consider the same problem for the Lipschitz observable βA. We investigate then the asymptotic limit when β\to \infty of the effective potential (which depends now on β) as well as the above family of probabilities. We relate the limit objects with an ergodic version of Kantorovich transshipment problem. In statistical mechanics β\propto 1/T, where T is the absolute temperature. In this way, we are also analyzing the problem related to the effective potential at temperature zero.

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arXiv2010

Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_λ(p,v)=\frac{1}{2}g_{p}(v,v)- λω_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $ω$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-ω_{p}(v)$ has support in a unique closed geodesic $γ$; and the curvature is negative at every point of $M$ but at the points of $γ$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $γ$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.

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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.

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arXiv2010

Quantum Stochastic Processes, Quantum Iterated Function Systems and Entropy

We describe some basic results for Quantum Stochastic Processes and present some new results about a certain class of processes which are associated to Quantum Iterated Function Systems (QIFS). We discuss questions related to the Markov property and we present a definition of entropy which is induced by a QIFS. This definition is a natural generalization of the Shannon-Kolmogorov entropy from Ergodic Theory. This definition is different from the one in the paper "A Thermodynamic Formalism for density matrices in Quantum Information" by the same authors.

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arXiv2010

A Thermodynamic Formalism for density matrices in Quantum Information

We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the definitions and results we describe here are natural generalizations of the classical concepts of Thermodynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is well-known that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism. We consider the operator $Λ$ acting on the space of density matrices $\mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $$ Λ(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)\frac{V_iρV_i^*}{tr(V_iρV_i^*)}, $$ where $W_i$ and $V_i$, $i=1,2,..., k$ are linear operators in this Hilbert space. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i\,(.)\,V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (i.e., the dynamics on the configuration space of density matrices) and the $W_i$ play the role of the weights one can consider on the IFS. In this way a family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS). We also present some estimates related to the Holevo bound.

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arXiv2010

Zeta measures and Thermodynamic Formalism for temperature zero

We address the analysis of the following problem: given a real Hölder potential $f$ defined on the Bernoulli space and $μ_f$ its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a Hölder function $f>0$ and a value $s$ such that $0 0$. We do not assume here the maximizing probability for $f$ is unique.

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arXiv2009

On generic $G$-prevalent properties of $C^{r}$ diffeomorphisms of $\mathbf{S}^{1}$ and a quantitative K-S theorem

We will consider a convex unbounded set and certain group of actions $G$ on this set. This will substitute the translation (by adding) structure usually consider in the classical setting of prevalence. In this way we will be able to define the meaning of $G$-prevalent set. In this setting we will show a kind of quantitative Kupka-Smale Theorem and also a result about rotation numbers which was first consider by J.-C. Yoccoz (and, also by M. Tsujii).

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arXiv2009

On calibrated and separating sub-actions

Consider a transitive expanding dynamical system $ σ: Σ\to Σ$, and a Hölder potential $ A $. In ergodic optimization, one is interested in properties of $A$-maximizing probabilities. Assuming ergodicity, it is already known that the projection of the support of such probabilities is contained in the set of non-wandering points with respect to $ A $, denoted by $ Ω(A) $. A separating sub-action is a sub-action such that the sub-cohomological equation becomes an identity just on $ Ω(A) $. For a fixed Hölder potential $ A $, we prove not only that there exists Hölder separating sub-actions but in fact that they define a residual subset of the Hölder sub-actions. We use the existence of such separating sub-actions in an application for the case one has more than one maximizing probability. Suppose we have a finite number of distinct $A$-maximizing probabilities with ergodic property: $ \hat μ_j $, $ j \in \{1, 2, ..., l\} $. Considering a calibrated sub-action $ u $, under certain conditions, we will show that it can be written in the form $$ u (\mathbf x)= u (\mathbf x^i) + h_A(\mathbf x^i, \mathbf x), $$ for all $ \mathbf x \in Σ$, where $ \mathbf x^i $ is a special point (in the projection of the support of a certain $ \hat μ_i $) and $ h_A $ is the Peierls barrier associated to $ A $.

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arXiv2009

Minimax probabilities for Aubry-Mather Problems

In this paper we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First we give some examples that show that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are $T$-periodic minimax Mather measures. We give a complete characterization of these measures and discuss several examples.

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arXiv2009

Negative Entropy, Zero temperature and stationary Markov Chains on the interval

We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^\mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^\mathbb{N}\to \mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $μ_\infty$ on $ [0,1]^\mathbb{N}$ that maximize the value $ \int A d μ,$ among all stationary Markov probabilities $μ$ on $[0,1]^\mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $μ_β$ which weakly converges to $μ_\infty$. The probabilities $μ_β$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $(x,y) \in [0,1]^2$, we show the graph property.

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arXiv2008

KMS States, Entropy and a Variational Principle for Pressure

We want to relate the concepts of entropy and pressure to that of KMS states for $C^*$-Algebras. Several different definitions of entropy are known in our days. The one we describe here is quite natural and extends the usual one for Dynamical Systems in Thermodynamic Formalism Theory. It has the advantage of been very easy to be introduced. It is basically obtained from transfer operators (also called Ruelle operators). Later we introduce the concept of pressure as a min-max principle. Finally, we consider the concept of a KMS state as an equilibrium state for a potential (in the context of $C^*$-Algebras) and we show that there is a relation between KMS states for certain algebras coming from a continuous transformation and equilibrium measures.

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