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Arnaldo Nogueira

Publications and source records attributed to Arnaldo Nogueira.

At least 19 recordsLinked to original sources
arXiv2023

On the complexity of polygonal billiards

We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same sequence of sides as the orbit of an aperiodic phase point z converges to z. We establishe a polynomial lower bound estimate on this convergence rate for almost every z. This yields an upper bound on the upper metric complexity and upper slow entropy of polygonal billiards. We also prove significant deviations from the expected convergence behavior. Finally we extend these results to higher dimensions as well as to arbitrary invariant measures.

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arXiv2022

Rotation number of 2-interval piecewise affine maps

We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $f_{\p}$ are parametrized by a quintuple $\p$ of real numbers satisfying inequations. Viewing $f_{\p}$ as a circle map, we show that it has a rotation number $ρ(f_{\p})$ and we compute $ρ(f_{\p})$ as a function of $\p$ in terms of Hecke-Mahler series. As a corollary, we prove that $ρ(f_{\p})$ is a rational number when the components of $\p$ are algebraic numbers.

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arXiv2021

Dynamics of piecewise increasing contractions

Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$ admits at most $k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear $λ$-affine maps, where $0<λ<1$. Let $b_1,\ldots,b_k$ be real numbers and let $F_λ: I\to \mathbb{R}$ be a function such that each restriction $F_λ|_{I_i}(x)=λx +b_i$. Under a generic assumption on the parameters $a_1,\ldots,a_{k-1},b_1,\ldots,b_k$, we prove that, up to a zero Hausdorff dimension set of slopes $λ$, the $ω$-limit set of the piecewise $λ$-affine maps $f_λ:x\in I \mapsto F_λ(x)\pmod{1}$ at every point equals a periodic orbit and there exist at most $k$ periodic orbits. Moreover, let $\mathfrak{E}^{(k)}$ be the exceptional set of parameters $λ,a_1,\ldots,a_{k-1},b_1,\ldots,b_k$ which define non-asymptotically periodic $f$, we prove that $\mathfrak{E}^{(k)}$ is a Lebesgue null measure set whose Hausdorff dimension is large or equal to $k$.

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arXiv2020

Chess Billiards

We show the chess billiard map, which was introduced in [HM] in order to study a generalization of the $n$-Queens problem in chess, is a circle homeomorphism. We give a survey of some of the known results on circle homeomorphisms, and apply them to this map. We prove a number of new results which give answers to some of the open questions posed in [HM].

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arXiv2019

Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $δ$ and $ϕ$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.

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arXiv2019

Diophantine approximation with nonsingular integral transformations

Let $Γ$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\leq p \leq n-1$ and $V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise action of $Γ$ on $V$. Let $\bx\in V$ be such that $Γ\bx$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $\by \in V$ by the orbit $\{ γ\bx \mid γ\in Γ\}$, in terms of $\norm γ\norm$, and prove in particular that for all $\bx$ in the complement of a specific null set described in terms of a certain Diophantine condition, the exponent of approximation is $(n-p)/p$; that is, for any $ρ<(n-p)/p$, $\norm γ\bx - \by \norm < \norm γ\norm^{-ρ}$ for infinitely many $γ$.

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arXiv2018

Rotation number of interval contracted rotations

Let $0<λ<1$. We consider the one-parameter family of circle $λ$-affine contractions $f_δ:x \in [0,1) \mapsto λx + δ\; {\rm mod}\,1 $, where $0 \le δ<1$. Let $ρ$ be the rotation number of the map $f_δ$. We will give some numerical relations between the values of $λ,δ$ and $ρ$, essentially using Hecke-Mahler series and a tree structure. When both parameters $λ$ and $δ$ are algebraic numbers, we show that $ρ$ is a rational number. Moreover, in the case $λ$ and $δ$ are rational, we give an explicit upper bound for the height of $ρ$ under an assumption on $λ$.

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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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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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arXiv2013

Classical homogeneous multidimensional continued fraction algorithms are ergodic

Homogeneous continued fraction algorithms are multidimensional generalizations of the classical Euclidean algorithm, the dissipative map $$ (x_1,x_2) \in \mathbb{R}_+^2 \longmapsto \left\{\begin{array}{ll} (x_1 - x_2, x_2), & \mbox{if $x_1 \geq x_2$} (x_1, x_2 - x_1), & \mbox{otherwise.} \end{array} \right. $$ We focus on those which act piecewise linearly on finitely many copies of positive cones which we call Rauzy induction type algorithms. In particular, a variation Selmer algorithm belongs to this class. We prove that Rauzy induction type algorithms, as well as Selmer algorithms, are ergodic with respect to Lebesgue measure.

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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

Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transfomations

The two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, as Jacobi-Perron, Poincaré, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.

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arXiv2011

Absorbing sets of homogeneous subtractive algorithms

We consider homogeneous multidimensional continued fraction algorithms, in particular a family of maps which was introduced by F. Schweiger. We prove his conjecture regarding the existence of an absorbing set for those maps. We also establish that their renormalisations are nonergodic which disproves another conjecture due to Schweiger. Other homogeneous algorithms are also analysed including ones which are ergodic.

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arXiv2011

Orbit structure of interval exchange transformations with flip

A sharp bound on the number of invariant components of an interval exchange transformation is provided. More precisely, it is proved that the number of periodic components n_per and the number of minimal components n_min of an interval exchange transformation of n intervals satisfy n_per+2 n_min\le n. Besides, it is shown that almost all interval exchange transformations are typical, that is, have all the periodic components stable and all the minimal components robust (i.e. persistent under almost all small perturbations). Finally, we find all the possible values for the integer vector (n_per, n_min) for all typical interval exchange transformation of n intervals.

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