arXiv subjects

Benito Pires

Publications and source records attributed to Benito Pires.

At least 19 recordsLinked to original sources

arXiv2025

Multidimensional contracted rotations

We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in \textit{Acta Arithmetica} in 1999. More precisely, we show that if $A$ is an invertible linear contraction of $\mathbb{R}^d$, then the map $f: [0,1)^d\to [0,1)^d$ defined by $f(x) = Ax +b\,\,(\textrm{mod}\,\mathbb{Z}^d)$ is asymptotically periodic for Lebesgue almost all $b\in\mathbb{R}^d$. We also include an example of a family of multidimensional contracted rotations $(d>1)$ not conjugate to the product of one-dimensional contracted rotations $(d=1)$, showing that our result cannot be reduced to or derived from the one-dimensional result of Bugeaud and Conze.

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arXiv2024

Multi-dimensional piecewise contractions are asymptotically periodic

Piecewise contractions (PCs) are piecewise smooth maps that decrease distance between pairs of points in the same domain of continuity. The dynamics of a variety of systems is described by PCs. During the last decade, a lot of effort has been devoted to proving that in parametrized families of one-dimensional PCs, the $\omega$-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor $\omega$-limit sets. In this article, we extend these results to the multi-dimensional case. More precisely, we provide criteria to show that an arbitrary family $\{f_{\mu}\}_{\mu\in U}$ of locally bi-Lipschitz piecewise contractions $f_\mu:X\to X$ defined on a compact metric space $X$ is asymptotically periodic for Lebesgue almost every parameter $\mu$ running over an open subset $U$ of the $M$-dimensional Euclidean space $\mathbb{R}^M$. As a corollary of our results, we prove that piecewise affine contractions of $\mathbb{R}^d$ defined in generic polyhedral partitions are asymptotically periodic.

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arXiv2023

Chaotic time series in financial processes consisting of savings with piecewise constant monthly contributions

We investigate the time series generated by an elementary and deterministic financial process that consists in making monthly contributions to a savings account subjected to the devaluation by a monthly negative real interest rate. The monthly contribution is a piecewise constant function of the account balance. We show that a dichotomy holds for such a financial time series: either the financial time series are asymptotic to finitely many periodic sequences or the financial time series have an uncountable (Cantor) set of ω-limit points. We also provide explicit parameters for which the financial process is chaotic in the sense that the financial time series have sensitive dependence on initial conditions at points of a Cantor attractor.

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arXiv2022

Multiple colour interacting urns on complete graphs

We present a multiple colour generalisation of the model of graph interacting urns studied by Benaim et. al., Random Struct. Alg., 46: 614-634, 2015. We show that for complete graphs and for a broad class of reinforcement functions governing the addition of balls in the urns, the process of colour proportions at each urn converges almost surely to the fixed points of the reinforcement function.

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arXiv2022

A chaotic discrete-time continuous-state Hopfield network with piecewise-affine activation functions

We construct a chaotic discrete-time continuous-state Hopfield network with piecewise-affine nonnegative activation functions and weight matrix with small positive entries. More precisely, there exists a Cantor set $C$ in the state space such that the network has sensitive dependence on initial conditions at initial states in $C$ and the network orbit of each initial state in $C$ has $C$ as its $ω$-limit set. The approach we use is based on tools developed and employed recently in the study of the topological dynamics of piecewise-contractions. The parameters of the chaotic network are explicitly given.

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arXiv2021

Switched server systems whose parameters are normal numbers in base 4

Switched server systems are mathematical models of manufacturing, traffic and queueing systems. Recently, it was proved in (Eur. J. Appl. Math. 31(4) (2020), pp. 682-708) that there exist switched server systems with $3$ buffers (tanks), a server, filling rates $ρ_1=ρ_2=ρ_3=\frac13$ and parameters $d_1, d_2, d_3>0$ whose global attractor is a fractal set. In this article, we prove that if $x_1$ in $(0,\frac13)$, $x_2$ in $(\frac13,\frac23)$ and $x_3$ in $(\frac23,1)$ are rational numbers or normal numbers in base $4$ (or more generally, rich numbers to base $4$) and $(d_1,d_2,d_3)$ is the vector with positive entries satisfying $$d_1=\frac{1}{3x_1}-1,\quad d_2=\frac{2-3x_2}{3x_2-1}, \quad d_3=\frac{3-3x_3}{3x_3-2},$$ then the corresponding switched server has no fractal attractor. More precisely, the Poincaré map of the system has a finite global attractor. The approach we use is to study the topological dynamics of a family of piecewise $λ$-affine contractions that includes the Poincaré map of the switched server system as a particular case.

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arXiv2021

Chaos and frequent hypercyclicity for composition operators

The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Rusza in 2015 that for backward weighted shifts on $\ell_p(\mathbb{Z})$, the notions chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^p$-spaces the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible frequently hypercyclic operator on $\ell_1$ whose inverse is not frequently hypercyclic is constructed.

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arXiv2020

Vertex reinforced random walks with exponential interaction on complete graphs

We describe a model for $m$ vertex reinforced interacting random walks on complete graphs with $d\geq 2$ vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if the absolute value of the interaction strength parameters are smaller than a certain upper bound, then, for any number of random walks ($m\geq 2$) on any graph ($d \geq 2$), the vertex occupation measure converges toward a unique equilibrium. We provide two additional examples of repelling random walks for the cases $m=d=2$ and $m=3$, $d=2$. The latter is used to study some properties of three exponentially repelling random walks on $\mathbb{Z}$.

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arXiv2019

Piecewise contractions and b-adic expansions

Let $I=[0,1)$, $b\in \{2,3,\ldots\}$ and $f:I\to I$ be an injective piecewise $\frac{1}{b}$-affine map, that is, assume that there exists a partition of $I$ into intervals $I_1,\ldots,I_n$ such that $\vert f(x)-f(y)\vert\le\frac1b \vert x-y\vert$ for all $x,y\in I_i$ and $1\le i\le n$. In this note, we study the $δ$-parameter family of maps $f_δ=R_δ\circ f$, where $R_δ:x\mapsto \{x+δ\}$. More precisely, we show that the set $\mathcal{N}$ of parameters $δ$ for which $f_δ$ has only natural codings with maximal complexity is a non-empty set with Hausdorff \mbox{dimension $0$}. We also show that for all $δ\in\mathcal{N}$, the map $f_δ$ is topologically semiconjugate to a minimal $n$-interval exchange transformation satisfying Keane's i.d.o.c. condition. The main result turns out to be a concrete application of the result by Mauduit and Moreira that the set of numbers having $b$-adic expansion with entropy $0$ has Hausdorff dimension $0$.

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arXiv2019

A switched server system semi-conjugate to a minimal interval exchange

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behavior: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations with flips. The ultimate result is a switched server system whose Poincare map is semiconjugate to a minimal and uniquely ergodic interval exchange transformation with flips.

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arXiv2018

Symbolic dynamics of piecewise contractions

A map $f{:}\,[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals} ($n$-PC) if there exist $0<λ<1$ and a partition of $[0,1)$ into intervals $I_1,\ldots,I_n$ such that $f\vert_{I_i}$ is $λ$-Lipschitz for every $1\le i\le n$. An infinite word $θ=θ_0θ_1\ldots$ over the alphabet $\mathcal{A}=\{1,\ldots,n\}$ is a {\it natural coding of} $f$ if there exists $x\in I$ such that $θ_k=i$ if and only if $f^k(x)\in I_i$. We prove that if $θ$ is a natural coding of an injective $n$-PC, then some infinite subword of $θ$ is either periodic or isomorphic to a natural coding of a topologically transitive $m$-interval exchange transformation ($m$-IET), where $m\le n$. Conversely, every natural coding of a topologically transitive $n$-IET is also a natural coding of some injective $n$-PC.

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arXiv2017

Topological transivity and mixing of the composition operators

Let $X=(X,\mathcal{B},μ)$ be a $σ$-finite measure space and \mbox{$f:X\to X$} be a measurable transformation such that the composition operator $T_f:φ\mapsto φ\circ f$ is a bounded linear operator acting on $L^p(X,\mathcal{B},μ)$, $1\le p<\infty$. We provide a necessary and sufficient condition on $f$ for $T_f$ to be topologically transitive or topologically mixing. We also characterize the topological dynamics of composition operators induced by weighted shifts, non-singular odometers and inner functions. The results provided in this article hold for composition operators acting on more general Banach spaces of functions.

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arXiv2016

Topological dynamics of piecewise λ-affine maps

Let $-1<λ<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $λ$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\ldots,b_n$ such that $f(x)=λx+b_i$ for every $x\in [c_{i-1},c_i)$. We prove that, for Lebesgue almost every $δ\in\mathbb{R}$, the map $f_δ=f+δ\,({\rm mod}\,1)$ is asymptotically periodic. More precisely, $f_δ$ has at most $2n$ periodic orbits and the $ω$-limit set of every $x\in [0,1)$ is a periodic orbit.

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arXiv2016

Invariant measures for piecewise continuous maps

We say that $f:[0,1]\to [0,1]$ is a {\it piecewise continuous interval map} if there exists a partition $0=x_0<x_1<\cdots<x_{d}<x_{d+1}=1$ of $[0,1]$ such that $f\vert_{(x_{i-1},x_i)}$ is continuous and the lateral limits $w_0^+=\lim_{x\to 0^+} f(x)$, $w_{d+1}^-=\lim_{x\to 1^-} f(x)$, \mbox{$w_i^{-}=\lim_{x\to x_i^-} f(x)$} and $w_i^{+}=\lim_{x\to x_i^+} f(x)$ exist for each $i$. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semi-conjugate to an interval exchange transformation.

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arXiv2014

Piecewise contractions defined by iterated function systems

Let $ϕ_1,\ldots,ϕ_n:[0,1]\to (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\cdots <x_{n-1}<1$, the piecewise contraction $f:I\to I$ defined by $x\in [x_{i-1},x_i)\mapsto ϕ_i(x)$ is asymptotically periodic. More precisely, $f$ has at least one and at most $n$ periodic orbits and the $ω$-limit set $ω_f(x)$ is a periodic orbit of $f$ for every $x\in I$.

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arXiv2013

Asymptotically periodic piecewise contractions of the interval

We consider the iterates of a generic injective piecewise contraction of the interval defined by a finite family of contractions. Let $ϕ_i:[0,1]\to (0,1)$, $1\le i\le n$, be $C^2$-diffeomorphisms with $\sup_{x\in (0,1)} \vert Dϕ_i(x)\vert<1$ whose images $ϕ_1([0,1]), \ldots, ϕ_n([0,1])$ are pairwise disjoint. Let $0<x_1<\cdots<x_{n-1}<1$ and let $I_1,\ldots, I_n$ be a partition of the interval $[0,1)$ into subintervals $I_i$ having interior $(x_{i-1},x_i)$, where $x_0=0$ and $x_n=1$. Let $f_{x_1,\ldots,x_{n-1}}$ be the map given by $x\mapsto ϕ_i(x)$ if $x\in I_i$, for $1\le i\le n$. Among other results we prove that for Lebesgue almost every $(x_1,\ldots,x_{n-1})$, the piecewise contraction $f_{x_1,\ldots,x_{n-1}}$ is asymptotically periodic.

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arXiv2012

Dynamics of piecewise contractions of the interval

We study the asymptotical behaviour of iterates of piecewise contractive maps of the interval. It is known that Poincaré first return maps induced by some Cherry flows on transverse intervals are, up to topological conjugacy, piecewise contractions. These maps also appear in discretely controlled dynamical systems, describing the time evolution of manufacturing process adopting some decision-making policies. An injective map $f:[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals}, if there exists a partition of the interval $[0,1)$ into $n$ intervals $I_1$,..., $I_n$ such that for every $i\in{1,...,n}$, the restriction $f|_{I_i}$ is $κ$-Lipschitz for some $κ\in (0,1)$. We prove that every piecewise contraction $f$ of $n$ intervals has at most $n$ periodic orbits. Moreover, we show that every piecewise contraction is topologically conjugate to a piecewise linear contraction.

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arXiv2011

On global linearization of planar involutions

Let $ϕ:\R^2\to\R^2$ be an orientation--preserving $C^1$ involution such that $ϕ(0)=0$ and let ${\rm Spc}\,(ϕ)=\{{\rm Eigenvalues\,\,of}\,\, Dϕ(p)\mid p\in\R^2\}$. We prove that if ${\rm Spc}\,{(ϕ)}\subset\R$ or ${\rm Spc}\,(ϕ)\cap [1,1+ε)=\emptyset$ for some $ε>0$ then $ϕ$ is globally $C^1$ conjugate to the linear involution $Dϕ(0)$ via the conjugacy $h=(I+Dϕ(0)ϕ)/2$, where $I:\R^2\to\R^2$ is the identity map. Similarly, if $ϕ$ is an orientation-reversing $C^1$ involution such that $ϕ(0)=0$ and ${\rm Trace}\,\big(Dϕ(0)Dϕ(p)\big)>-1 $ for all $p\in\R^2$ then $ϕ$ is globally $C^1$ conjugate to the linear involution $Dϕ(0)$ via the conjugacy $h$. Finally, we show that $h$ may fail to be a global linearization of $ϕ$ if the above conditions are not fulfilled.

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