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Julien Melleray

Publications and source records attributed to Julien Melleray.

At least 19 recordsLinked to original sources
arXiv2026

Some aspects of topological dynamics of Polish groups (with an introduction to descriptive set theory)

The first part of these notes give an introduction to the theory of Polish group actions on compact Hausdorff spaces, leading up to a proof of the Kechris-Pestov-Todorcevic correspondence and discussions of properties of universal minimal flows. The second part proveides some background on descriptive set theory and culminates with B. Miller's proof of the $\mathcal{G}_0$-dichotomy theorem due to Kechris, Solecki, and Todorcevic.

math.LO↗
arXiv2025

Dense and comeager conjugacy classes in zero-dimensional dynamics

Given a countable group $G$, we initiate a systematic study of the Polish spaces of all minimal and topologically transitive actions of $G$ on the Cantor space by homeomorphisms, with a focus on the existence of comeager conjugacy classes in these spaces. We develop a general model-theoretic framework to study this and related questions, recovering on the way many existing results from the literature. A substantial part of the paper is devoted to actions of free groups. We show that in that case, there is a comeager conjugacy class in the space of minimal actions, as well as in the space of minimal, probability measure-preserving actions. The first one is the Fra\"iss\'e limit of all sofic minimal subshifts and the second, the universal profinite action. The case of the integers was already treated by Hochman and there the two actions coincide with the universal odometer. In the non-abelian case, they are substantially different and new techniques are required. In the opposite direction, if $G$ is an amenable group which is not finitely generated, we show that there is no comeager conjugacy class in the space of all actions, and if $G$ is locally finite, also in the space of minimal actions. Finally, we study the question of existence of a dense conjugacy class in the space of topologically transitive actions. We show that if $G$ is hyperbolic or virtually polycyclic, then such a dense conjugacy class exists iff $G$ is virtually cyclic, suggesting that the case of the integers may be exceptional.

math.DS↗
arXiv2025

A dynamical proof of Matui's absorption theorem

We give a dynamical, relatively elementary proof of an "absorption theorem" which is closely related to a well-known result due to Matui. The construction is in the spirit of an earlier joint work of the author and S. Robert. In an appendix we explain how to use this result to correct the dynamical proof given by Melleray--Robert of a classification theorem for orbit equivalence of minimal ample groups due to Giordano, Putnam and Skau (the original argument had a gap).

math.DS↗
arXiv2023

From invariant measures to orbit equivalence, via locally finite groups

We give a new proof of a theorem of Giordano, Putnam and Skau characterizing orbit equivalence of minimal homeomorphisms of the Cantor space in terms of their sets of invariant Borel probability measures. The proof is based on a strengtehning of a theorem of Krieger concerning minimal actions of certain locally finite groups of homeomorphisms, and we also give a new proof of the Giordano--Putnam--Skau characterization of orbit equivalence for these actions.

math.DS↗
arXiv2023

A topometric Effros theorem

Given a continuous and isometric action of a Polish group $G$ on an adequate Polish topometric space $(X,τ,ρ)$ and $x \in X$, we find a necessary and sufficient condition for $\overline{Gx}^ρ$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.

math.GN↗
arXiv2022

Clopen type semigroups of actions on $0$-dimensional compact spaces

We investigate some properties of the clopen type semigroup of an action of a countable group on a compact, $0$-dimensional, Hausdorff space X. We discuss some characterizations of dynamical comparison (most of which were already known in the metrizable case) in this setting; and prove that for a Cantor minimal action $\alpha$ of an amenable group the topological full group of $\alpha$ admits a dense, locally finite subgroup iff the corresponding clopen type semigroup is unperforated. We also discuss some properties of clopen type semigroups of the Stone-\v{C}ech compactifications and universal minimal flows of countable groups, and derive some consequences on generic properties in the space of minimal actions of a given countable group on the Cantor space.

math.DS↗
arXiv2021

Generic properties of homeomorphisms preserving a given dynamical simplex

Given a dynamical simplex $K$ on a Cantor space $X$, we consider the set $G_K^*$ of all homeomorphisms of $X$ which preserve all elements of $K$ and have no nontrivial clopen invariant subset. Generalising a theorem of Yingst, we prove that for a generic element $g$ of $G_K^*$ the set of invariant measures of $g$ is equal to $K$. We also investigate when there exists a generic conjugacy class in $G_K^*$ and prove that this happens exactly when $K$ has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer is generic in $G_K^*$.

math.DS↗
arXiv2021

Homogeneous actions on Urysohn spaces

We show that many countable groups acting on trees, including free products of infinite countable groups and surface groups, are isomorphic to dense subgroups of isometry groups of bounded Urysohn spaces. This extends previous results of the first and last author with Y. Stalder on dense subgroups of the automorphism group of the random graph. In the unbounded case, we also show that every free product of infinite countable groups arises as a dense subgroup of the isometry group of the rational Urysohn space.

math.GR↗
arXiv2020

Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic countable universal locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn $Δ$-metric spaces in terms of the distance value set $Δ$.

math.LO↗
arXiv2018

Polish groups with metrizable universal minimal flows

We prove that if the universal minimal flow of a Polish group $G$ is metrizable and contains a $G_δ$ orbit $G \cdot x_0$, then it is isomorphic to the completion of the homogeneous space $G/G_{x_0}$ and show how this result translates naturally in terms of structural Ramsey theory. We also investigate universal minimal proximal flows and describe concrete representations of them in a number of examples.

math.DS↗
arXiv2018

Dynamical simplices and Borel complexity of orbit equivalence

We prove that any divisible dynamical simplex is the set of invariant measures of some Toeplitz subshift. We apply our construction to prove that orbit equivalence of Toeplitz subshifts is Borel bireducible to the universal equivalence relation induced by a Borel action of a nonarchimedean Polish group.

math.LO↗
arXiv2018

Dynamical simplices and Fraïssé theory

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$. We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz' theorem stating that any Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of D. Ash.

math.DS↗
arXiv2016

Dynamical simplices and minimal homeomorphisms

We give a characterization of sets K of probability measures on a Cantor space X with the property that there exists a minimal homeomorphism g of X such that the set of g-invariant probability measures on X coincides with K. This extends theorems of Akin (corresponding to the case when K is a singleton) and Dahl (when K is finite-dimensional). Our argument is elementary and different from both Akin's and Dahl's.

math.DS↗
arXiv2016

From isolated subgroups to generic permutation representations

Let $G$ be a countable group, $\operatorname{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chabauty topology and $\operatorname{Is}(G) \subset \operatorname{Sub}(G)$ the collection of isolated points. We denote by $X!$ the (Polish) group of all permutations of a countable set $X$. Then the following properties are equivalent: (i) $\operatorname{Is}(G)$ is dense in $\operatorname{Sub}(G)$, (ii) $G$ admits a "generic permutation representation". Namely there exists some $τ^* \in \operatorname{Hom}(G,X!)$ such that the collection of permutation representations $\{ϕ\in \operatorname{Hom}(G,X!) \ | \ ϕ{\text{is permutation isomorphic to}} τ^*\}$ is co-meager in $\operatorname{Hom}(G,X!)$. We call groups satisfying these properties solitary. Examples of solitary groups include finitely generated LERF groups and groups with countably many subgroups.

math.GR↗
arXiv2015

Isometrisable group actions

Given a separable metrisable space X, and a group G of homeomorphisms of X, we introduce a topological property of the action of G on X which is equivalent to the existence of a G-invariant compatible metric on X. This extends a result of Marjanovic obtained under the additional assumption that X is locally compact.

math.LO↗
arXiv2014

Grey subsets of Polish spaces

We develop the basics of an analogue of descriptive set theory for functions on a Polish space $X$. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

math.LO↗
arXiv2014

Extremely amenable groups via continuous logic

We establish a characterization of extreme amenability of any Polish group in Fraïssé-theoretic terms in the setting of continuous logic, mirroring a theorem due to Kechris, Pestov and Todorcevic for closed subgroups of the permutation group of an infinite countable set.

math.LO↗