arXiv subjects
Publications and source records attributed to Elismar R. Oliveira.
We present algorithms to compute approximations of invariant measures and its attractors for IFS and GIFS, using the deterministic algorithm in a tractable way, with code optimization strategies and use of data structures and search algorithms. The results show that these algorithms allow the use of these (G)IFS in a reasonable running time.
We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ., 10 (2018), 1, 172--178. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as greyscale images.
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.
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 $1$.
We show that the number of Laplacian eigenvalues greater than the average degree of a tree having $n$ vertices is at most $\lfloor\frac{n}{2} \rfloor$.
We address the problem of ordering trees with the same degree sequence by their spectral radii. To achieve that, we consider 2-switch transformations which preserve the degree sequence and establish when the index decreases. Our main contribution is to determine a total ordering of a particular family by their indices according to a given parameter related to sizes in the tree.
We introduce the discrete version of the Hutchinson--Barnsley theory providing algorithms to approximate the Hutchinson measure for iterated function systems (IFS) and generalized iterated function systems (GIFS) complementing the discrete version of the deterministic algorithm considered in our previous work DOS to generate attractors of both classical and fuzzy IFS and GIFS.
We provide a new algorithm to generate images of the generalized fuzzy fractal attractors described in Oliveira-2017. We also provide some important results on the approximation of fractal operators to discrete subspaces with application to discrete versions of the deterministic algorithm for fractal image generation in the cases of IFS recovering the classical images from Barnsley et al., Fuzzy IFS from Cabrelli-1992 and GIFS's from Jaros-2016.
We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behavior and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing measures and the critical value through a discounting limit. This shows the connection between this maximization problem and the superior boundary of the compact invariant set, which is described as a graph of the solution of the Bellman equation.
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.
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.
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.
The index of a graph is the largest eigenvalue of its adjacency matrix. A starlike is a tree having a unique vertex of degree $r>2$. We show how to order the starlike trees with $n>3$ by their indices. In particular, the indices of starlike trees are all distinct.
For, $0<λ<1$, consider the transformation $T(x) = d x $ (mod 1) on the circle $S^1$, a $C^1$ function $A:S^1 \to \mathbb{R}$, and, the map $F(x,s) = ( T(x) , λ\, s + A(x))$, $(x,s)\in S^1 \times \mathbb{R}$. We denote $\mathcal{B}= \mathcal{B}_λ$ the upper boundary of the attractor (known as fat attractor). We are interested in the regularity of $\mathcal{B}_λ$, and, also in what happens in the limit when $λ\to 1$. We also address the analysis of the following conjecture which were proposed by R. Bamón, J. Kiwi, J. Rivera-Letelier and R. Urzúa: for any fixed $λ$, $C^1$ generically on the potential $A$, the upper boundary $\mathcal{B}_λ$ is formed by a finite number of pieces of smooth unstable manifolds of periodic orbits for $F$. We show the proof of the conjecture for the class of $C^2$ potentials $A(x)$ satisfying the twist condition (plus a combinatorial condition). We do not need the generic hypothesis for this result. We present explicit examples. On the other hand, when $λ$ is close to $1$ and the potential $A$ is generic a more precise description can be done. In this case the finite number of pieces of $C^1$ curves on the boundary have some special properties. Having a finite number of pieces on this boundary is an important issue in a problem related to semi-classical limits and micro-support. This was consider in a recent published work by A. Lopes and J. Mohr. Finally, we present the general analysis of the case where $A$ is Lipschitz and its relation with Ergodic Transport
Cabrelli, Forte, Molter and Vrscay in 1992 considered a {fuzzy} version of the theory of iterated function systems (IFSs in short) and their fractals%The idea was to extend the classical Hutchinson-Barnsley operator to selfmaps of a metric space to appropriate selfmaps of space of fuzzy , which now is quite rich and important part of the fractals theory. On the other hand, Miculescu and Mihail in 2008 introduced another generalization of the IFSs' theory - instead of selfmaps of a metric space $X$, they considered mappings defined on the finite Cartesian product $X^m$. %It turns out that many parts of the classical Hutchinson-Barnsley fractals theory have natural counterparts in this generalized setting. In particular, if $X$ is complete, then appropriately contractive systems of such maps generate unique fractal sets. In this paper we show that the \emph{fuzzyfication} ideas of Cabrelli et al. can be naturally adjusted to the case of mappings defined on finite Cartesian product. In particular, we define the notion of a generalized iterated fuzzy function system (GIFZS in short) and prove that it generates a unique fuzzy fractal set. We also study some basic properties of GIFZSs and their fractals, and consider the question whether our setting gives us some new fuzzy fractal sets.
In 1987, J. H. Elton, has proved the first fundamental result in convergence of IFS, the Elton's Ergodic Theorem. In this work we prove the natural extension of this theorem to the projected Hutchinson measure $μ_α$ associated to a GIFSpdp $\mathcal{S}=\left(X, (ϕ_j:X^{m} \to X)_{j=0,1, ..., n-1}, (p_j)_{j=0,1, ..., n-1}\right),$ in a compact metric space $(X,d)$. More precisely, the average along of the trajectories $x_{n}(a)$ of the GIFS, starting in any initial points $x_0, ..., x_{m-1} \in X$ satisfies, for any $f \in C(X , \mathbb{R})$, $$\lim_{N\to +\infty} \frac{1}{N}\sum_{n=0 }^{N-1} f(x_{n}(a)) = \int_{X} f(t) dμ_α(t),$$ for almost all $a \in Ω=\{0,1, ..., n-1\}^{\mathbb{N}}$, the symbolic space. Additionally, we give some examples and applications to Chaos Games and Nonautonomous Dynamical Systems defined by finite difference equations.
For $X$, $Y$, $Z$ and $W$ compact metric spaces, consider two uniformly contractive IFS $\{τ_x: Z\to Z,\, x\in x\}$ and $\{τ_y:W\to W,\, y\in Y\}$. For a fixed $α\in \mathcal{P}(X)$ with $supp(α)=X$ we define the entropy of a holonomic measure $π\in \mathcal{P}(X\times Z)$ relative to $α$, the pressure of a continuous cost function $c(x,z)$ and show that for $c$ Lipschitz this pressure coincides with the spectral radius of the associated transfer operator. The same approach can be applied to the pair $Y,W$. For fixed probabilities $α\in \mathcal{P}(X)$ and $β\in \mathcal{P}(Y)$ with $supp(α)=X,\,supp(β)=Y$ we denote by $H_α(π), π\in Π(\cdot,\cdot,τ)$, the entropy of the $(X,Z)-$marginal of $π$ relative to $α$ and denote by $H_β(π)$, the entropy of the $(Y,W)-$marginal of $π$ relative to $β$. The marginal pressure of a continuous cost function $c \in C(X\times Y \times Z \times W)$ relative to $(α,β)$ will be defined by $P^{m}(c) = \sup_{π\inΠ(\cdot,\cdot,τ)} \int c\, dπ+ H_α(π) +H_β(π)$ and we will show the following duality result: \[\inf_{P^{m}(c -φ(x) -ψ(y))=0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν,τ)} \int c\, dπ+ H_α(π) +H_β(π).\] When $Z$ and $W$ have only one point and the entropy is unconsidered this equality can be rewritten as the Kantorovich Duality for compact spaces $X,Y$ and continuous cost $-c$: \[\inf_{c -φ(x) -ψ(y)\leq 0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν)} \int c\, dπ.\]
Consider the shift $σ$ acting on the Bernoulli space $Σ={1,2,...,n}^\mathbb{N}$. We denote $\hatΣ= {1,2,...,n}^\mathbb{Z}$. We analyze several properties of the maximizing probability $μ_{\infty,A}$ of a Holder potential $A: Σ\to \mathbb{R}$. Associated to $A(x)$, via the involution kernel, $W: \hatΣ \to \mathbb{R}$, it is known that can we get the dual potential $A^*(y)$, where $(x,y)\in \hatΣ$. Consider $μ_{\infty, A^*}$ a maximizing probability for $A^*$. We would like to consider the transport problem from $μ_{\infty,A}$ to $μ_{\infty,A^*}$. In this case, it is natural to consider the cost function $c(x,y) = I(x) - W(x,y) +γ$, where $I$ is the deviation function. The pair of functions for the Kantorovich Transport dual Problem are $(-V,-V^*$), where we denote the two calibrated sub-actions by $V$ and $V^*$, respectively, for $A$ and $A^*$ for $μ_{\infty,A}$. We analyze the graph property for the optimal plan $\hatμ$.