arXiv subjects

Artur O. Lopes

Publications and source records attributed to Artur O. Lopes.

81 records · Page 5Linked to original sources

arXiv2008

The Mather measure and a Large Deviation Principle for the Entropy Penalized Method

We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $μ$ is unique and the support of $μ$ is the Aubry set). Consider, for each value of $ε$ and h, the entropy penalized Mather problem $\min \{\int_{\tn\times\rn} L(x,v)dμ(x,v)+εS[μ]\},$ where the entropy S is given by $S[μ]=\int_{\tn\times\rn}μ(x,v)\ln\frac{μ(x,v)}{\int_{\rn}μ(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $μ(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure $μ_{ε, h}$ which converges to the Mather measure $μ$. We show a LDP $\lim_{ε,h\to0} ε\ln μ_{ε,h}(A),$ where $A\subset \mathbb{T}^N\times\mathbb{R}^N$. The deviation function I is given by $I(x,v)= L(x,v)+\nablaϕ_0(x)(v)-\bar{H}_{0},$ where $ϕ_0$ is the unique viscosity solution for L.

math.DS↗

arXiv2008

Eigenfunctions of the Laplacian and associated Ruelle operator

Let $Γ$ be a co-compact Fuchsian group of isometries on the Poincaré disk $\DD$ and $Δ$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $Δ$, equivariant by $Γ$ with real eigenvalue $λ=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution $\dd_{f,s}$ (the Helgason distribution) which is equivariant by $Γ$ and supported at infinity $\partial\DD=\SS^1$. The geodesic flow on the compact surface $\DD/Γ$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:\SS^1\to\SS^1$, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that $\dd_{f,s}$ is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $ψ_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \[ ψ_{f,s} (ξ)=\int \frac{J(ξ,η)}{|ξ-η|^{2s}} \dd_{f,s} (dη), \] \noindent where $J(ξ,η)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $\DD/Γ$.

math.DS↗

arXiv2008

Poncelet pairs and the Twist Map associated to the Poncelet Billiard

We show that for a fixed curve $K$ and for a family of variables curves $L$, the number of $n$-Poncelet pairs is $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd $ (m,n)=1$. The curvee $K$ do not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion .

math.DS↗

arXiv2008

Renormalization for a Class of Dynamical Systems: some Local and Global Properties

We study the period doubling renormalization operator for dynamics which present two coupled laminar regimes with two weakly expanding fixed points. We focus our analysis on the potential point of view, meaning we want to solve $$V=\mathcal{R} (V):=V\circ f\circ h+V \circ h,$$ where $f$ and $h$ are naturally defined. Under certain hypothesis we show the existence of a explicit ``attracting'' fixed point $V^*$ for $\mathcal{R} $. We call $\mathcal{R}$ the renormalization operator which acts on potentials $V$. The log of the derivative of the main branch of the Manneville-Pomeau map appears as a special ``attracting'' fixed point for the local doubling period renormalization operator. We also consider an analogous definition for the one-sided 2-full shift $§$ (and also for the two-sided shift) and we obtain a similar result. Then, we consider global properties and we prove two rigidity results. Up to some weak assumptions, we get the uniqueness for the renormalization operator in the shift. In the last section we show (via a certain continuous fraction expansion) a natural relation of the two settings: shift acting on the Bernoulli space $\{0,1\}^\mathbb{N}$ and Manneville-Pomeau-like map acting on an interval.

math.DS↗

arXiv2007

Entropy and Variational principles for holonomic probabilities of IFS

Associated to a IFS one can consider a continuous map $\hatσ : [0,1]\times Σ\to [0,1]\times Σ$, defined by $\hatσ(x,w)=(τ_{X_{1}(w)}(x), σ(w))$ were $Σ=\{0,1, ..., d-1\}^{\mathbb{N}}$, $σ: Σ\to Σ$ is given by$σ(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : Σ\to \{0,1, ..., n-1\}$ is the projection on the coordinate $k$. A $ρ$-weighted system, $ρ\geq 0$, is a weighted system $([0,1], τ_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] \to \mathbb{R}$ and probability $ν$ on $[0,1]$ satisfying $ P_{u}(h)=ρh, \quad P_{u}^{*}(ν)=ρν$. A probability $\hatν$ on $[0,1]\times Σ$ is called holonomic for $\hatσ$ if $ \int g \circ \hatσ d\hatν= \int g d\hatν, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by ${\cal H}$. Via disintegration, holonomic probabilities $\hatν$ on $[0,1]\times Σ$ are naturally associated to a $ρ$-weighted system. More precisely, there exist a probability $ν$ on $[0,1]$ and $u_i, i\in\{0, 1,2,..,d-1\}$ on $[0,1]$, such that is $P_{u}^*(ν)=ν$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $ϕ\in \mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(ϕ)=$$

math.DS↗

arXiv2007

Parameter Estimation in Manneville-Pomeau Processes

In this work we study a class of stochastic processes $\{X_t\}_{t\in\N}$, where $X_t = (ϕ\circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability μ_s on [0,1] and a continuous by part function $ϕ:[0,1] \to \R$. We consider here $T_s:[0,1]\to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, on the smoothed periodogram function, on the variance of the partial sum and on the wavelet theory.

math.ST↗

arXiv2007

On the Aubry-Mather theory for symbolic dynamics

We propose a new model of ergodic optimization for expansive dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In another contexts (for twist maps, for instance), this property appears in a crucial way. A version of the Aubry-Mather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the two-sided shift. Under the transitive hypothesis, we show the existence of sub-actions for Holder potentials also in the holonomic setting. We analyze then connections between calibrated sub-actions and the Mane potential. A representation formula for calibrated sub-actions is presented, which drives us naturally to a classification theorem for these sub-actions. We also investigate properties of the support of maximizing probabilities.

math.DS↗

arXiv2007

Exponential decay of correlation for the Stochastic Process associated to the Entropy Penalized Method

In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let $L(x, v):\Tt^n\times\Rr^n\to \Rr$ be a Lagrangian of the form L(x,v) = {1/2}|v|^2 - U(x) + < P, v>. For each value of $ε$ and $h$, consider the operator \Gg[ϕ](x):= -εh {ln}[\int_{\re^N} e ^{-\frac{hL(x,v)+ϕ(x+hv)}{εh}}dv], as well as the reversed operator \bar \Gg[ϕ](x):= -εh {ln}[\int_{\re^N} e^{-\frac{hL(x+hv,-v)+ϕ(x+hv)}{εh}}dv], both acting on continuous functions $ϕ:\Tt^n\to \Rr$. Denote by $ϕ_{ε,h} $ the solution of $\Gg[ϕ_{ε,h}]=ϕ_{ε,h}+λ_{ε,h}$, and by $\bar ϕ_{ε,h} $ the solution of $\bar \Gg[ϕ_{ε,h}]=\bar ϕ_{ε,h}+λ_{ε,h}$. In order to analyze the decay of correlation for this process we show that the operator $ {\cal L} (ϕ) (x) = \int e^{- \frac{h L (x,v)}ε} ϕ(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum.

math.DS↗

arXiv2006

Functions for relative maximization

We introduce functions for relative maximization in a general context: the beta and alpha applications. After a systematic study concerning regularities, we investigate how to approximate certain values of these functions using periodic orbits. We establish yet that the differential of an alpha application dictates the asymptotic behavior of the optimal trajectories.

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