We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to λ-approximate conjugacy, where λ is the infinite measure which is preserved. This sharpens the classical version of the Rokhlin lemma, which only provides such a classification up to μ-approximate conjugacy where μ is a probability measure equivalent to λ.
Research
-
On the Rokhlin lemma for infinite measure-preserving bijections (with François Le Maître),
Preprint. arXiv, pdf.
ABSTRACT -
Polish full groups preserving an infinite measure,
Submitted. arXiv, pdf.
ABSTRACTWe extend some results of Carderi and Le Maître on full groups in the probability context to the infinite measure one: there exists at most one Polish group topology (refining the weak topology and coarser than the uniform topology) on an ergodic full group, and the orbit full group of a locally compact group acting in a Borel manner can be endowed with a Polish group topology. Moreover, orbit full groups are complete invariants for orbit equivalence. We then generalize a result from Le Maître on non-Polishability of the group of finitely supported bijections: the finitely supported elements of an ergodic full group carries a Polish group topology if and only if the associated full group comes from a countable equivalence relation. We finish with algebraic and topological results on ergodic (orbit) full groups concerning normal subgroups, contractibility and genericity of aperiodic elements.
-
Spatial models for boolean actions in the infinite measure-preserving setup (with François Le Maître),
Transactions of the AMS. arXiv, pdf, DOI.
ABSTRACTWe show that up to a null set, every infinite measure-preserving action of a locally compact Polish group can be turned into a continuous measure-preserving action on a locally compact Polish space where the underlying measure is Radon. We also investigate the distinction between spatial and boolean actions in the infinite measure-preserving setup. We finally obtain a streamlined proof of a recent result of Avraham- Re’em and Roy: Lévy groups cannot admit nontrivial continuous measure-preserving actions on Polish spaces when the measure is locally finite.
-
On measure-preserving actions of Polish groups on spaces of infinite measure, my PhD thesis.
Here is the pdf version, and here are the slides of the defense.
Talks
-
25/03/2026: Orsay PhD student seminar, Laboratoire de Mathématiques d'Orsay, Orsay.
ABSTRACTIn this talk I will present the basic context of measured dynamics, and more specifically of ergodic theory on standard measured spaces. After redefining standard measure spaces, measure-preserving bijections and ergodicity, the focus of my talk will be to expose some examples. The odometer is a very classical example of an ergodic bijection preserving a probability measure, but obtaining examples of infinite measure-preserving ergodic bijections (IMPEB) is a "non-trivial task". I will use two different representations of the odometer to present two general classes of IMPEB, the Kakutani skyscrapers and the infinite Rank-one bijections. While the definitions will be a bit informal to avoid drowning in technical details, I will integrate as many drawings as I can to give the intuition behind the constructions. If time allows, I will give some properties that these classes of IMPEB satisfy, in order to explain why they are so central in the theory, and/or give some sketches of proofs.
-
22/05/2025: Bourbakette, IMJ-PRG, Paris.
ABSTRACT(Bourbakettes are informal talks aimed at PhD students, and their goal is to present interesting math in an accessible way.) The game of SET! is a game about finding affine lines in (Z/3Z)^4. The so-called "capset" problem is about finding maximal sets in (Z/3Z)^4 that do not contain such lines. We will start with a succinct presentation of a general result of Meshulam from 1995 giving an upper bound for the cardinal of such a set in a general abelian group. We will then go back to (Z/3Z)^d (with exact results this time), and spend some time with the cases d=2 and d=3 in order to understand the ideas for d=4. Based on a M2 research project.
-
22/05/2025: IMJ-PRG PhD seminar, IMJ-PRG, Paris.
ABSTRACTThis talk will be centered on an invariant of orbit equivalence introduced by H. Dye in 1959, the full group. In a first time I will define this object and give a few basic examples, then I will briefly mention the theorem that motivates their study: the reconstruction theorem. This theorem encourages us to view full groups as topological groups, therefore in a second time I will talk about certain natural topologies that we can consider. The main theorem of this part states that in some (nice) cases we can always find a Polish topology on full groups preserving an infinite measure. Finally, if time allows, I will discuss some topological and/or algebraic properties, and I will insist on the differences with the case of a full group perserving a probability measure.
-
17/04/2025: Seminar Algebra–Geometrie–Kombinatorik, Technische Universität Dresden.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action, an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. After presenting all the relevant notions, I will present a result from a joint work with François Le Maître, in which we show that it is possible to obtain such a model for actions of locally compact Polish groups.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
12/02/2025: LAREMA PhD seminar, Université d'Angers, Angers.
ABSTRACTDans cet exposé je parlerai d'un invariant d'équivalence orbitale introduit par H. Dye en 1959, le groupe plein. Dans un premier temps je définirai cet objet et je donnerai quelques exemples, puis j'énoncerai le théorème principal qui en motive l'étude, le théorème de reconstruction. Ce théorème encourage l'étude de ces groupes pleins en tant que groupes topologiques, et ainsi dans un deuxième temps je parlerai de certaines topologies naturelles sur les groupes pleins préservant une mesure infinie, le théorème principal de cette partie affirmant que dans certains cas on peut toujours trouver une topologie Polonaise. Enfin, si le temps le permet, je discuterai de certaines propriétés topologiques, en insistant sur les différences avec le cas d'une mesure de probabilité.
-
10/01/2025: Groupe de travail de théorie ergodique, Université Paris Cité, Paris.
ABSTRACTDye's reconstruction theorem states that any abstract isomorphism between two ergodic full groups on a standard probability space is actually a conjugation by some measure-preserving bijection. This is useful for instance for proving that full groups are complete invariants of orbit equivalence. When working with infinite measure or with non-singular bijections however, we need a more general version of the theorem, which was later proved by Fremlin in the language of Boolean algebras. I will give some details about his proof, which relies on an algebraic description of the involutions in the full group, and will work specifically in the context of measure algebras. See Notes.
-
13/12/2024: Arbre de Noël 2024 en Géométrie Non Commutative, Orléans, France.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action, an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of lcsc groups.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
11/04/2024: Séminaire d'Algèbres d'Opérateurs, IMJ-PRG, Paris.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action: an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of locally compact groups, while spatial actions of a certain class of large groups can only be trivial.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
08/04/2024: Séminaire Groupes et actions, Université Paris-Saclay, Orsay.
ABSTRACTAny measure-preserving Borel action of a Polish group on a standard measured space -called a spatial action- gives a measure-preserving boolean action, an action on the Borel subsets up to measure zero. Conversely, a natural problem is the following: given a boolean action, does it come from a spatial action? Glasner, Tsirelson and Weiss gave a complete answer for finite measures. Moreover, when a spatial action exists, a theorem of Becker and Kechris implies that we can always take it to be a continuous action on a compact space. For the case of infinite measures however, we need to take into account the interplay between measure and topology, and to this end we aim to obtain a continuous action on a locally compact Polish space endowed with a Radon measure. In a joint work with François Le Maître, we show that it is possible to obtain such a model for actions of locally compact groups, while spatial actions of a certain class of large groups can only be trivial.
About the article: Spatial models for boolean actions in the infinite measure-preserving setup. See Research. -
30/05/2023: Groupe de travail de théorie ergodique, Université Paris Cité, Paris.
ABSTRACTThis talk is about a proof of Belinskaya's theorem, stating that given an ergodic measure-preserving transformation of a standard probability space, any other transformation with the same orbits and an integrable cocycle must be flip-conjugate to it. The proof is due to Katznelson and available in the appendix of the paper: "Belinskaya's theorem is optimal", by Carderi, Joseph, Le Maître and Tessera. See Notes.
Notes
-
Some notes about the construction of some Examples of ergodic infinite measure-preserving bijections.
ABSTRACTThis very informal note aims to present different ∞-mp ergodic bijections on standard σ-finite spaces. Constructing examples of such bijections is ‘a non-trivial task’ according to Halmos, but since Kakutani’s skyscraper construction in 1943, which to our knowledge is the first significant and non-trivial one, many other examples have been studied. We go into a bit more details and provide some proofs in the first sections about Kakutani skyscrapers, as they represent the most relevant examples for the author’s work, and remain more concise for the other sections, with some exceptions.
-
Some notes about the Becker and Kechris theorem on Polish G-space realization.
ABSTRACTIn this very informal note we recall some basic statements of descriptive set theory, and use them, along with some more specific notions which are also recalled, to give two proofs of the theorem of Becker and Kechris. This theorem allows one to endow a Borel G-space (for G Polish) with a compatible topology making it a Polish G-space. The original proof, and a refinement from Hjorth, are given.
-
A short tutorial about using Git and VS Code, tailor-made for people in a very specific situation.
ABSTRACTA short tutorial made for people under Windows 10, not used to working with Git. It contains basic help for installing Git, Visual Studio Code, how to create a SSH key and how to clone your projects specifically from plmlab to your local folders. It is destined to people who do not need to understand everything, but just need easy access to a way to cooperate on a project.
-
Some notes about Fremlin's generalization of Dye's reconstruction theorem.
ABSTRACTIn this note we provide context and give a demonstration of a generalization of Dye’s reconstruction theorem, proved by Fremlin, following Eigen’s arguments. The theorem is stated for the very general setting of Dedekind complete Boolean algebras and their automorphism groups, but we will specifically work with measure algebras of standard spaces.
-
Some handwritten notes about Belinskaya's theorem (in french), for the ergodic theory working group.
ABSTRACTThese notes provide a detailed proof of Belinskaya's theorem (two ergodic pmp bijections of a standard probability space that are orbit equivalent are flip conjugate as soon as one of the cocycles is integrabe), due to Katznelson, and published in this article by Alessandro Carderi, Matthieu Joseph, François Le Maître and Romain Tessera.
-
My Master thesis on Property (T) for measured groupoids (in french), supervised by François Le Maître.
ABSTRACTThis master thesis is based on an article from Claire Anantharaman-Delaroche, and more specifically about a theorem on Kazhdan's property (T) for measured groupoids. The necessary background about groupoids, measure disintegration, unitary representations and affine Hilbert spaces is introduced, then the main theorem is stated and proved. A direct consequence is the following: if G is a countable group acting in a free, pmp and ergodic manner on a standard probability space, then the associated equivalence relation has property (T) if and only if G has property (T). Some comments are made at the end about an extension of the theorem to the locally compact setting.
Teaching
-
2026-2027 (first semester) at Université Paris Cité
- Exercise sessions for "Groups and group actions" (for 3rd year students).
- Tutoring sessions for 1st, 2nd and 3rd year students.
- 2023-2026 Colles (Regular oral examinations) for students of ECG1 Applied Mathematics, Lycée Alexandre Dumas, Saint-Cloud.
- 2021-2026 High school level Mathematics courses, Lycée Alexandre Dumas, Saint-Cloud.
- 2020-2021 High school level Mathematics courses, Lycée Alain, Le Vésinet.
- 2019-2020 Middle school level Mathematics courses, Collège Jean Rostand, Nîmes.
- 2018 Agrégation externe de Mathématiques.