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Jérôme Los

Publications and source records attributed to Jérôme Los.

8 recordsLinked to original sources
arXiv2025

A group from a map and orbit equivalence

In two papers published in 1979, R. Bowen and C. Series defined a dynamical system from a Fuchsian group, acting on the hyperbolic plane $\mathbb{H}^2$. The dynamics is given by a map on $S^1$ which is, in particular, an expanding piecewise homeomorphism of the circle. In this paper we consider a reverse question: which dynamical conditions for an expanding piecewise homeomorphism of $S^1$ are sufficient for the map to be a ``Bowen-Series-type" map (see below) for some group $G$ and which groups can occur? We give a partial answer to these questions.

math.DS↗
arXiv2022

Geometrization of Piecewise homeomorphisms of the circle

In this work we study the following realization problem: given a piecewise homeomorphism $Φ: S^1 \rightarrow S^1$, which geometrical and dynamical conditions on $Φ$ are sufficient to realize it as a Bowen-Series-like map associated to a surface group, seen as a discrete subgroup of ${\rm Homeo}^+ (S^1)$?

math.DS↗
arXiv2016

A note on covers of fibred hyperbolic manifolds

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $φ_1$ and $φ_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $φ_1$ to $\tilde S$ with respect to $p_1$ coincides with that of $φ_2$ with respect to $p_2$. The mapping tori of the $φ_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways.

math.GT↗
arXiv2014

Volume entropy for minimal presentations of surface groups in all ranks

We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called \emph{minimal geometric presentations}. We rediscover a formula first obtained by Cannon and Wagreich with the computation in a non published manuscript by Cannon. The result is surprising: an explicit polynomial of degree $n$, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely different. It is based on a dynamical system construction following an idea due to Bowen and Series and extended to all geometric presentations in. The result is an explicit formula for the volume entropy of minimal presentations for all surface groups, showing a polynomial dependence in the rank $n>2$. We prove that for a surface group $G_n$ of rank $n$ with a classical presentation $P_n$ the volume entropy is $\log(λ_n)$, where $λ_n$ is the unique real root larger than one of the polynomial \[ x^{n} - 2(n - 1) \sum_{j=1}^{n-1} x^{j} + 1. \]

math.DS↗
arXiv2009

Volume entropy for surface groups via Bowen-Series like maps

We define a Bowen-Series like map for every geometric presentation of a co-compact surface group and we prove that the volume entropy of the presentation is the topological entropy of this particular (circle) map. Finally we find the minimal volume entropy among geometric presentations.

math.DS↗
arXiv2009

Infinite sequence of fixed point free pseudo-Anosov homeomorphisms

We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy-Veech induction strategy.

math.DS↗
arXiv2007

Roots, symmetries and conjugacy of pseudo-Anosov mapping classes

An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem.

math.DS↗