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Arnaud Hilion

Publications and source records attributed to Arnaud Hilion.

At least 19 recordsLinked to original sources
arXiv2025

PolExp growth for automorphisms of toral relatively hyperbolic groups

Let $G$ be a toral relatively hyperbolic group, and let $\varphi\in\mathrm{Aut}(G)$. We prove that, under iteration of $\varphi$, the conjugacy length $||\varphi^n(g)||$ of every element $g\in G$ grows like $n^d\lambda^n$ for some $d\in\mathbb{N}$ and some algebraic integer $\lambda\geq 1$. For a given $\varphi$, only finitely many values of $d$ and $\lambda$ occur as $g$ varies in $G$. The same statements hold for the growth of the word length $|\varphi^n(g)|$. For $G$ hyperbolic, we generalize polynomial subgroups: we show that, for a given growth type $n^d\lambda^n$ other than $1$, there is a malnormal family of quasiconvex subgroups $K_1,\dots,K_p$ such that a conjugacy class $[g]$ grows at most like $n^d\lambda^n$ if and only if $g$ is conjugate into one of the subgroups $K_i$.

math.GR↗
arXiv2022

Measure transfer and $S$-adic developments for subshifts

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$.

math.DS↗
arXiv2022

The measure transfer for subshifts induced by a morphism of free monoids

Every non-erasing monoid morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $\sigma_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(\sigma(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(\sigma(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $\sigma(X) \subset \mathcal{B}^{\mathbb{Z}}$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure $\sigma_X^{\mathcal{M}(\mu)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $\sigma^{\mathcal{M}_X}$ is injective. \smallskip The assumption on $\sigma$ that it is ``injective on the shift-orbits of $X$'' is strictly weaker than ``recognizable in $X$'', and strictly stronger than ``recognizable for aperiodic points in $X$''. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map $\sigma_X^{\mathcal{M}}$.

math.DS↗
arXiv2022

A Pansiot-type subword complexity theorem for automorphisms of free groups

Inspired by Pansiot's work on substitutions, we prove a similar theorem for automorphisms of a free group F of finite rank: if a right-infinite word X represents an attracting fixed point of an automorphism of F, the subword complexity of X is equivalent to n, n log log n, n log n, or n^2. The proof uses combinatorial arguments analogue to Pansiot's as well as train tracks. We also define the recurrence complexity of X, and we apply it to laminations. In particular, we show that attracting laminations have complexity equivalent to n, n log log n, n log n, or n^2 (to n if the automorphism is fully irreducible).

math.GR↗
arXiv2020

Invariant measures on finite rank subshifts

In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.

math.DS↗
arXiv2019

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow Γ$ and vector towers $\overleftarrow v$ on $\overleftarrow Γ$, which allows us to efficiently describe all invariant measures $μ= μ^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $φ: F_N \to F_N$ is determined, when $φ$ is represented by a train track map.

math.DS↗
arXiv2019

Tower power for $S$-adics

We explain and restate the results from our recent paper arXiv:1503.08000.v3 in standard language for substitutions and $S$-adic systems in symbolic dynamics. We then produce as rather direct application an $S$-adic system (with finite set of substitutions $S$ on $d$ letters) that is minimal and has $d$ distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure $\mu([w])$, for any word $w \in \cal A^*$ and any invariant measure $\mu$ on the subshift $X_\sigma$ defined by any everywhere growing but not necessarily primitive or irreducible substitution $\sigma: \cal A^* \to \cal A^*$. Several examples are considered in detail, and model computations are presented.

math.DS↗
arXiv2018

The visual boundary of hyperbolic free-by-cyclic groups

Let $ϕ$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_ϕ = F_n \rtimes_ϕ \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite graph $K_{3,3}$ into $\partial G_ϕ$. To do so, we define the directional Whitehead graph and prove that an indecomposable $F_n$-tree is Levitt type if and only if one of its directional Whitehead graphs contains more than one edge. As an application, we obtain a direct proof of Kapovich-Kleiner's theorem that $\partial G_ϕ$ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.

math.GT↗
arXiv2017

Topological substitutions and Rauzy fractals

We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.

math.DS↗
arXiv2014

Ergodic currents dual to a real tree

Let $T$ be an $\R$-tree in the boundary of Outer space with dense orbits. When the free group $\FN$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\R$-tree $T$. Given a current $μ$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards $μ$. We also give a unique ergodicity criterion.

math.GR↗
arXiv2013

The hyperbolicity of the sphere complex via surgery paths

Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's sphere complex (another model for the free splitting complex), instead of Handel and Mosher's fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex. We explain how to deduce from our proof the hyperbolicity of some other complexes such as the free factor complex or the arc complex (of a surface with boundary).

math.GT↗
arXiv2012

Geometric realizations of two dimensional substitutive tilings

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H}^2$. However, we give an example of substitutive tiling of $\Hyp^2$ which is non-primitive.

math.GT↗
arXiv2012

Rips Induction: Index of the dual lamination of an $\R$-tree

Let $T$ be a $\R$-tree in the boundary of the Outer Space CV$_N$, with dense orbits. The $Q$-index of $T$ is defined by means of the dual lamination of $T$. It is a generalisation of the Euler-Poincaré index of a foliation on a surface. We prove that the $Q$-index of $T$ is bounded above by $2N-2$, and we study the case of equality. The main tool is to develop the Rips Machine in order to deal with systems of isometries on compact $\R$-trees. Combining our results on the $\CQ$-index with results on the classical geometric index of a tree, we obtain a beginning of classification of trees. As a consequence, we give a classification of iwip outer automorphisms of the free group, by discussing the properties of their attracting and repelling trees.

math.GR↗
arXiv2011

Indecomposable $F_N$-trees and minimal laminations

We extend the techniques of [CH] to build an inductive procedure for studying actions in the boundary of the Culler-Vogtmann Outer Space, the main novelty being an adaptation of he classical Rauzy-Veech induction for studying actions of surface type. As an application, we prove that a tree in the boundary of Outer space is free and indecomposable if and only if its dual lamination is minimal up to diagonal leaves. Our main result generalizes [BFH97, Proposition 1.8] as well as the main result of [KL11].

math.GR↗
arXiv2009

$\R$-trees, dual laminations, and compact systems of partial isometries

Let $\FN$ be a free group of finite rank $N \geq 2$, and let $T$ be an $\R$-tree with a very small, minimal action of $\FN$ with dense orbits. For any basis $\CA$ of $\FN$ there exists a {\em heart} $K_{\CA} \subset \bar T$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}$, for each $a_{i} \in \CA$, defines a tree $T_{(K_{\CA}, \CA)}$ which contains an isometric copy of $T$ as minimal subtree.

math.GR↗