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Emmanuel Militon

Publications and source records attributed to Emmanuel Militon.

16 recordsLinked to original sources
arXiv2026

Distortion in the group of locally monotone homeomorphisms of a Cantor set and in the group of generalized interval exchange transformations

Let f be either a generalized interval exchange transformation or a locally monotone homeomorphism of a Cantor subset of the real line. In this article, we prove that the following are equivalent. 1. The number of discontinuities of f^n is bounded. 2. There exists n $\ge$ 1 such that the element f is conjugate to the restriction to a closed invariant subset of a disjoint union of n circles of a homeomorphism of this disjoint union of circles. 3. The element f is distorted in the group of generalized interval exchange transformations or in the group of locally monotone homeomorphisms of the Cantor subset.

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arXiv2025

Almost reducibility, distortion and local perfection for diffeomorphisms of one-manifolds

In this article, we characterize the distortion elements of the group of smooth diffeomorphisms of the circle and of the group of compactly supported smooth diffeomorphisms of the real line. More precisely, we prove that, in this context, an element is distorted if and only if it is almost reducible, that is if and only if it has conjugates arbitrarily close to an isometry. For diffeomorphisms with fixed points, we show that this is equivalent to being the time-1 map of a C 1 vector field without hyperbolic zero. The equivalence between distortion and almost reducibility relies on new more general results about distortion elements in groups of diffeomorphisms of manifolds and on a new local perfection result for the group of compactly supported smooth diffeomorphisms of the real line.

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arXiv2023

Hyperbolic isometries of the fine curve graph of higher genus surfaces

We prove that for a homeomorphism f that is isotopic to the identity on a closed hyperbolic surface, the following are equivalent: * f acts hyperbolically on the fine curve graph; * f is isotopic to a pseudo-Anosov map relative to a finite f-invariant set; * the ergodic homological rotation set of f has nonempty interior.

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arXiv2023

Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative

In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probabilty measure on the circle or contains a free subgroup in two generators, which is reminiscent of the Tits alternatve for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of R.

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arXiv2023

Parabolic isometries of the fine curve graph of the torus

In this article we finish the classification of actions of torus homeomorphisms on the fine curve graph initiated by Bowden, Hensel, Mann, Militon, and Webb in \cite{BHMMW}. This is made by proving that if $f \in \mathrm{Homeo}(\mathbb{T}^2)$, then $f$ acts elliptically on $C^{\dagger}(\mathbb{T}^2)$ if and only if $f$ has bounded deviation from some $v \in \mathbb{Q}^2 \setminus \left\{0\right\}$. The proof involves some kind of slow rotation sets for torus homeomorphisms.

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arXiv2023

Groups of smooth diffeomorphisms of Cantor sets embedded in a line

Let K be a Cantor set embedded in the real line R. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of R. Higman-Thompson's groups Vn appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups Vn. We also prove that any finitely generated subgroup H without free subsemigroup on two generators is virtually abelian. The corresponding result for the groups Vn was unknown to our knowledge. As a consequence, those groups do not contain nilpotent groups which are not virtually abelian.

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arXiv2022

Homotopic rotation sets for higher genus surfaces

This paper states a definition of homotopic rotation set for higher genus surface homeomorphisms, as well as a collection of results that justify this definition. We first prove elementary results: we prove that this rotation set is star-shaped, we discuss the realisation of rotation vectors by orbits or periodic orbits and we prove the creation of new rotation vectors for some configurations.Then we use the theory developped by Le Calvez and Tal in [LCT18a] to obtain two deeper results:-- If the homotopical rotation set contains the direction of a closed geodesic which has a self-intersection, then there exists a rotational horseshoe and hence infinitely many periodic orbits in many directions.-- If the homotopical rotation set contains the directions of two closed geodesics that meet, there exists infinitely many periodic orbits in many directions.

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arXiv2021

Rotation sets and actions on curves

We study the action of the homeomorphism group of a surface $S$ on the fine curve graph ${\mathcal C }^\dagger(S)$. While the definition of $\mathcal{C}^\dagger(S)$ parallels the classical curve graph for mapping class groups, we show that the dynamics of the action of ${\mathrm{Homeo}}(S)$ on $\mathcal{C}^\dagger(S)$ is much richer: homeomorphisms induce parabolic isometries in addition to elliptics and hyperbolics, and all positive reals are realized as asymptotic translation lengths. When the surface $S$ is a torus, we relate the dynamics of the action of a homeomorphism on $\mathcal{C}^\dagger(S)$ to the dynamics of its action on the torus via the classical theory of rotation sets. We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.

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arXiv2017

Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms

Let S be a compact connected surface and let f be an element of the group Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d\_{n}/n) converges to 0, where d\_{n} is the diameter of \tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by B{é}guin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo\_{0}(S) is distorted if and only if it is non-spreading.

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arXiv2016

Distortion and Tits alternative in smooth mapping class groups

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting of diffeomorphisms which are isotopic to the identity on S does not contain any distorted elements. Moreover, we prove a weak Tits alternative for these groups.

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arXiv2013

Actions of groups of homeomorphisms on one-manifolds

In this article, we describe all the group morphisms from the group of compactly-supported homeomorphisms isotopic to the identity of a manifold to the group of homeomorphisms of the real line or of the circle.

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arXiv2013

Distortion elements for surface homeomorphisms

Let S be a compact orientable surface and f be an element of the group Homeo_{0}(S) of homeomorphisms of S isotopic to the identity. Denote by F a lift of f to the universal cover of S. In this article, the following result is proved: if there exists a fundamental domain D of the universal cover of S such that the sequence (d_{n}log(d_{n})/n) converges to 0 where d_{n} is the diameter of F^{n}(D), then the homeomorphism f is a distortion element of the group Homeo_{0}(S).

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arXiv2012

éléments de distorsion du groupe des difféomorphismes isotopes à l'identité d'une variété compacte

We consider, on a compact manifold, the group of diffeomorphisms that are isotopic to the identity. We show that every recurrent element is a distorsion element. This generalizes Avila's theorem on circle diffeomorphisms. The method also provides a new proof of a result by Calegari and Freedman: on a sphere, in the group of homeomorphisms that are isotopic to the identity, every element is distorted.

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arXiv2011

Commutator length of annulus diffeomorphisms

We study the group of C^{r}-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r different from 3, the linear space of homogeneous quasi-morphisms on this group is one dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphisms group is unbounded in the sense of Burago, Ivanov and Polterovich.

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