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Juliana Xavier

Publications and source records attributed to Juliana Xavier.

14 recordsLinked to original sources
arXiv2026

Non-locally connected spaces I: Coverings and branched coverings on metric spaces

Whether a rational function can have an indecomposable continuum as its Julia set is a fundamental open problem in complex dynamics. We settle this negatively for one of the most classical examples of an indecomposable continuum: we prove that the Knaster continuum cannot be the Julia set of any rational function. To obtain this result, we develop a theory of coverings and branched coverings on general (non-locally connected) metric spaces. We introduce a new Euler characteristic and extend the Riemann--Hurwitz formula to a wider class of spaces, including continua for which the classical shape-theoretic invariants are trivial.

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arXiv2022

One some planar Baumslag-Solitar actions

Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group for $n \geq 2$. We study representations of $BS(1, n)$ on the plane by orientation preserving homeomorphisms, assuming that $a$ acts as a linear map and $b$ as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map $h$ defined by the action of $a$. In case $h$ is diagonalizable over $\mathbb R$, we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases where $h$ is elliptic or parabolic. Then we give applications to the actions of $BS(1, n)$ by homeomorphisms of the torus.

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arXiv2022

Planar Baumslag-Solitar actions

Following previous work, where representations of $BS(1,n)$ by planar orientation preserving homeomorphisms with linear diagonalizable conjugating element was studied, we consider the elliptic and parabolic cases. As an application, we prove that there are no faithful representations of $BS(1,n)$ by toral homeomorphisms with conjugating element the Dehn twist map.

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arXiv2019

Linearization of topologically Anosov homeomorphisms of non compact surfaces

We study the dynamics of Topologically Anosov homeomorphisms of non compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f:S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \R ^ 2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in a previous work.

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arXiv2018

Topologically Anosov plane homeomorphisms

This paper deals with classifying the dynamics of {\it Topologically Anosov} plane homeomorphisms. We prove that a Topologically Anosov homeomorphism $f:\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwandering set of $f$ reduces to a fixed point, or if there exists an open, connected, simply connected proper subset $U$ such that $U \subset \mathrm{Int}(\overline {f(U)})$, and such that $ \cup_{n\geq 0} f^n (U)= \mathbb{R}^2$. In the general case, we prove a structure theorem for the $α$-limits of orbits with empty $ω$-limit (or the $ω$-limits of orbits with empty $α$-limit), and we show that any basin of attraction (or repulsion) must be unbounded.

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arXiv2016

Dynamics of covering maps of the annulus I: semiconjugacies

It is often the case that a covering map of the open annulus is semiconjugate to a map of the circle of the same degree. We investigate this possibility and its consequences on the dynamics. In particular, we address the problem of the classification up to conjugacy. However, there are examples which are not semiconjugate to a map of the circle, and this opens new questions.

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arXiv2016

Dynamics of annulus coverings II: periodic points

Let $f$ be a covering map of the open annulus $A= S^1\times (0,1)$ of degree $d$ , $|d|>1$. Assume that $f$ preserves an essential (i.e not contained in a disk of $A$) compact subset $K$. We show that $f$ has at least the same number of periodic points in each period as the map $z^d$ in $S^1.$

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arXiv2014

Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo-)Anosov elements

Let $BS(1,n)= $ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces with (pseudo-)Anosov elements. That is, we consider a closed surface $S$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is pseudo-Anosov with stretch factor $λ>1$. We show that $ $ is not a faithful representation of $BS(1, n)$ if $λ> n$. Moreover, we show that there are no faithful representations of $BS(1, n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $λ$).

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arXiv2012

A classification of minimal sets for surface homeomorphisms

We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by F. Kwakkel, T.Jäger and A. Passeggi in the torus. The given classification is studied in the non-wandering setting and in light of the Nielsen-Thurston Theory.

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arXiv2012

Surface Attractors

Let $f$ be a continuous endomorphism of a surface $M$, and $A$ an attracting set such that the restriction $f|_A: A \to A$ is a $d:1$ covering map. We show that if $f$ is a local homeomorphism in the immediate basin $B^0_A$ of $A$, then $f$ is also a $d:1$ covering of $B^0_A$.

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arXiv2012

Cycles of links and fixed points for orientation preserving homeomorphisms of the open unit disk

Michael Handel proved in Handel (1999) the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. More recently, the author generalized Handel's theorem to a wider class of cycles of links (Xavier 2012). In this paper we complete this topic describing exactly which are all the cycles of links forcing the existence of a fixed point.

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arXiv2012

Handel's fixed point theorem revisited

Michael Handel proved in [7] the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments [9]. These methods permitted to improve the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity

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