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E. Ugalde

Publications and source records attributed to E. Ugalde.

11 recordsLinked to original sources
arXiv2020

Gaussian concentration and uniqueness of equilibrium states in lattice systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space $S^{\mathbb{Z}^d}$ where $d\geq 1$ and $S$ is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

math.PR↗
arXiv2019

Gaussian Concentration bound for potentials satisfying Walters condition with subexponential continuity rates

We consider the full shift $T:Ω\toΩ$ where $Ω=A^{\mathbb N}$, $A$ being a finite alphabet. For a class of potentials which contains in particular potentials $ϕ$ with variation decreasing like $O(n^{-α})$ for some $α>2$, we prove that their corresponding equilibrium state $μ_ϕ$ satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant $C>0$ such that, for all $n$ and for all separately Lipschitz functions $K(x_0,\ldots,x_{n-1})$, the exponential moment of $K(x,\ldots,T^{n-1}x)-\int K(y,\ldots,T^{n-1}y)\, \mathrm{d}μ_ϕ(y)$ is bounded by $\exp\big(C\sum_{i=0}^{n-1}\mathrm{Lip}_i(K)^2\big)$. The crucial point is that $C$ is independent of $n$ and $K$. We then derive various consequences of this inequality. For instance, we obtain bounds on the fluctuations of the empirical frequency of blocks, the speed of convergence of the empirical measure, and speed of Markov approximation of $μ_ϕ$. We also derive an almost-sure central limit theorem.

math.DS↗
arXiv2014

On the Asymptotic Properties of Piecewise Contracting Maps

We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence and transitivity properties. Our examples include transitive and recurrent attractors, that are either finite, countable, or a disjoint union of a Cantor set and a countable set. We also show that the attractor of a piecewise contracting map is usually a Lebesgue measure-zero set, and we give conditions ensuring that it is totally disconnected. Finally, we provide an example of piecewise contracting map with positive topological entropy and whose attractor is an interval.

math.DS↗
arXiv2013

Symbolic Complexity for Nucleotide Sequences: A Sign of the Genome Structure

We introduce a method to estimate the complexity function of symbolic dynamical systems from a finite sequence of symbols. We test such complexity estimator on several symbolic dynamical systems whose complexity functions are known exactly. We use this technique to estimate the complexity function for genomes of several organisms under the assumption that a genome is a sequence produced by a (unknown) dynamical system. We show that the genome of several organisms share the property that their complexity functions behaves exponentially for words of small length $\ell$ ($0\leq \ell \leq 10$) and linearly for word lengths in the range $11 \leq \ell \leq 50$. It is also found that the species which are phylogenetically close each other have similar complexity functions calculated from a sample of their corresponding coding regions.

q-bio.PE↗
arXiv2011

Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

Let $A$ be a finite set and $ϕ:A^Z\to R$ be a locally constant potential. For each $β>0$ ("inverse temperature"), there is a unique Gibbs measure $μ_{βϕ}$. We prove that, as $β\to+\infty$, the family $(μ_{βϕ})_{β>0}$ converges (in weak-$^*$ topology) to a measure we characterize. It is concentrated on a certain subshift of finite type which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron-Frobenius Theorem for matrices á la Birkhoff. The crucial idea we bring is a "renormalization" procedure which explains convergence and provides a recursive algorithm to compute the weights of the ergodic decomposition of the limit.

math.DS↗
arXiv2011

On the finite-dimensional marginals of shift-invariant measures

Let $Σ$ be a finite alphabet, $Ω=Σ^{\mathbb{Z}^{d}}$ equipped with the shift action, and $\mathcal{I}$ the simplex of shift-invariant measures on $Ω$. We study the relation between the restriction $\mathcal{I}_n$ of $\mathcal{I}$ to the finite cubes $\{-n,...,n\}^d\subset\mathbb{Z}^d$, and the polytope of "locally invariant" measures $\mathcal{I}_n^{loc}$. We are especially interested in the geometry of the convex set $\mathcal{I}_n$ which turns out to be strikingly different when $d=1$ and when $d\geq 2$. A major role is played by shifts of finite type which are naturally identified with faces of $\mathcal{I}_n$, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of $\mathcal{I}_n$, although in dimension $d\geq 2$ there are also extreme points which arise in other ways. We show that $\mathcal{I}_n=\mathcal{I}_n^{loc}$ when $d=1$, but in higher dimension they differ for $n$ large enough. We also show that while in dimension one $\mathcal{I}_n$ are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of $\mathcal{I}_n$ for all large enough $n$.

math.DS↗
arXiv2004

Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources

Motivated by entropy estimation from chaotic time series, we provide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a $n$-cylinder fluctuates in the same way as the inverse measure of this $n$-cylinder at "small scales", but in a different way at "large scales". In particular, the Renyi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more difficult to handle theoretically. The results we obtain for return times, though less complete, improve the available ones.

math.DS↗
arXiv2004

Finite type approximations of Gibbs measures on sofic subshifts

Consider a Hölder continuous potential $ϕ$ defined on the full shift $A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $μ_ϕ$ on $X$ associated to $ϕ$. Besides, there is a natural nested sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift $X$. To this sequence we can associate a sequence of Gibbs measures $(μ_ϕ^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $μ_ϕ$ (in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of $μ_ϕ$. Finally, we prove that the measure-theoretic entropy of $μ_ϕ^m$ converges to the one of $μ_ϕ$ exponentially fast. We indicate how to extend our results to more general subshifts and potentials. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.

math.DS↗
arXiv2002

Projection of Markov measures may be Gibbsian

We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Holder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.

math.DS↗