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Frédéric Le Roux

Publications and source records attributed to Frédéric Le Roux.

At least 19 recordsLinked to original sources
arXiv2026

Approximating stable translation lengths on fine curve graphs

We study the stable translation length of homeomorphisms of a surface acting on the fine nonseparating curve graph and compare it to the stable translation lengths of its finite approximations - mapping classes relative to a finite invariant set - acting on the nonseparating curve graph. We prove that the stable translation length of a homeomorphism with a dense set of periodic points is the supremum of the stable translation lengths of its approximations, and that the stable translation length is preserved under cell-like extensions. We deduce that homotopically triv ial homeomorphisms of the torus have stable translation length which is the supremum of the stable translation lengths of their finite approximations. We show that the supremum is not always a maximum, by proving that the stable translation length of a mapping class acting on the nonseparating curve graph is rational.

math.DS↗
arXiv2025

Rotation sets and axes in the fine curve graph for torus homeomorphisms

We expand the dictionary between the action of a torus homeomorphism on the fine curve graph and its rotation set. More precisely, we show that the fixed points at infinity of a loxodromic element determine the rotation set up to scale. A key ingredient is a metric version of the classical WPD property from geometric group theory. As a consequence we find new stable criteria for positive scl, and for two homeomorphisms to generate a free group, and we provide a Tits alternative for groups of torus homeomorphisms.

math.GR↗
arXiv2022

Automorphisms of some variants of fine graphs

Recently Bowden, Hensel and Webb defined the fine curve graph for surfaces, extending the notion of curve graphs for the study of homeomorphism or diffeomorphism groups of surfaces. Later Long, Margalit, Pham, Verberne and Yao proved that for a closed surface of genus $g\geqslant 2$, the automorphism group of the fine graph is naturally isomorphic to the homeomorphism group of the surface. We extend this result to the torus case $g=1$; in fact our method works for more general surfaces, compact or not, orientable or not. We also discuss the case of a smooth version of the fine graph.

math.GT↗
arXiv2020

The Anosov-Katok method and pseudo-rotations in symplectic dynamics

We prove that toric symplectic manifolds admit Hamiltonian pseudo-rotations with a finite, and in a sense minimal, number of ergodic measures. The set of ergodic measures of these pseudo-rotations consists of the measure induced by the symplectic volume form and the Dirac measures supported at the fixed points of the torus action. Our construction relies on the conjugation method of Anosov and Katok.

math.SG↗
arXiv2018

Barcodes and area-preserving homeomorphisms

In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.

math.SG↗
arXiv2017

Polynomial entropy of Brouwer homeomorphisms

We study the polynomial entropy of the wandering part of any invertible dynamical system on a compact metric space. As an application we compute the polynomial entropy of Brouwer homeomorphisms (fixed point free orientation preserving homeomorphisms of the plane), and show in particular that it takes every real value greater or equal to 2.

math.DS↗
arXiv2017

Projective transformations of rotation sets

We give a new proof and extend a result of J. Kwapisz: whenever a set C is realized as the rotation set of some torus homeomorphism, the image of C under certain projective transformations is also realized as a rotations set.

math.DS↗
arXiv2017

Strong distortion in transformation groups

We discuss boundedness and distortion in transformation groups. We show that the groups $\mathrm{Diff}^r_0(\mathbb{R}^n)$ and $\mathrm{Diff}^r(\mathbb{R}^n)$ have the strong distortion property, whenever $0 \leq r \leq \infty, r \neq n+1$. This implies in particular that every abstract length function on these groups is bounded. With related techniques we show that, for $M$ a closed manifold or homeomorphic to the interior of a compact manifold with boundary, the groups $\mathrm{Diff}_0^r(M)$ satisfy a relative Higman embedding type property, introduced by Schreier. This answers a problem asked by Schreier in the famous Scottish Book.

math.DS↗
arXiv2017

Fixed point sets of isotopies on surfaces

We consider a self-homeomorphism h of some surface S. A subset F of the fixed point set of h is said to be unlinked if there is an isotopy from the identity to h that fixes every point of F. With Le Calvez' transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the two dimensional sphere that preserves the orientation and pointwise fix some given closed connected set F.

math.DS↗
arXiv2015

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces, we introduce a dynamical invariant, denoted by N, for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces. Along the way, we obtain several results of independent interest: We show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with N on autonomous Hamiltonians thus establishing a certain uniqueness result for spectral invariants, we obtain a "Max Formula" for spectral invariants on aspherical manifolds, give a very simple description of the Entov-Polterovich quasi-state on aspherical surfaces and characterize the heavy and super-heavy subsets of such surfaces.

math.SG↗
arXiv2012

An introduction to Handel's homotopy Brouwer theory

Homotopy Brouwer theory is a tool to study the dynamics of surface homeomorphisms. We introduce and illustrate the main objects of homotopy Brouwer theory, and provide a proof of Handel's fixed point theorem. These are the notes of a mini-course held during the workshop "Superficies en Montevideo" in March 2012.

math.DS↗
arXiv2011

Free planar actions of the Klein bottle group

We describe the structure of the free actions of the Klein bottle group by orientation preserving homeomorphisms of the plane. This group is generated by two elements $a,b$, where the conjugate of $b$ by $a$ equals the inverse of $b$. The main result is that $a$ must act properly discontinuously, while $b$ cannot act properly discontinuously. As a corollary, we describe some torsion free groups that cannot act freely on the plane. We also find some properties which are reminiscent of Brouwer theory for the infinite cyclic group $Z$, in particular that every free action is virtually wandering.

math.DS↗
arXiv2010

There is no minimal action of Z^2 on the plane

In this paper it is proved that there is no minimal action (i.e. every orbit is dense) of Z^2 on the plane. The proof uses the non-existence of minimal homeomorphisms on the infinite annulus (Le Calvez-Yoccoz's theorem), and the theory of Brouwer homeomorphisms.

math.DS↗
arXiv2009

Simplicity of the group of compactly supported area preserving homeomorphisms of the open disc and fragmentation of symplectic diffeomorphisms

In 1980, Albert Fathi asked whether the group of area-preserving homeomorphisms of the 2-disc that are the identity near the boundary is a simple group. In this paper, we show that the simplicity of this group is equivalent to the following fragmentation property in the group of compactly supported, area preserving diffeomorphisms of the plane: there exists a constant m such that every element supported on a disc D is the product of at most m elements supported on topological discs whose area are half the area of D.

math.DS↗
arXiv2008

Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett-Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.

math.DS↗