Given a Fell bundle $\mathscr C\overset {q}{\to }\Xi $ over the discrete groupoid $\Xi $, we study the symmetry of the associated Hahn algebra $\ell ^{\infty ,1}(\Xi \!\mid \!\mathscr C)$ in terms of the isotropy subgroups of $\Xi $. We prove that $\Xi $ is symmetric (respectively hypersymmetric) if and only if all of the isotropy subgroups are symmetric (respectively hypersymmetric). We also characterise hypersymmetry using Fell bundles with constant fibres, showing that for discrete groupoids, ‘hypersymmetry’ equals ‘rigid symmetry’.

Let G be a locally compact unimodular group, and let $\phi $ be some function of n variables on G. To such a $\phi $, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called “multiplicatively bounded $(p_1,\ldots ,p_n)$-norm” of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map $L_{p_1}(\mathbb {R}, S_{p_1}) \times L_{p_2}(\mathbb {R}, S_{p_2}) \rightarrow L_{1}(\mathbb {R}, S_{1})$, whenever $p_1$ and $p_2$ are such that $\frac {1}{p_1} + \frac {1}{p_2} = 1$. A similar result holds for certain Calderón–Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.

In this paper, we investigate the $H^{p}(G) \rightarrow L^{p}(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^{p}(G)$ is the Hardy space on $G$. Our main result extends those obtained in [Colloq. Math. 165 (2021), 1–30], where the $L^{1}(G)\rightarrow L^{1,\infty }(G)$ and $L^{p}(G) \rightarrow L^{p}(G)$, $1< p <\infty$, boundedness of such Fourier multiplier operators were proved.

We characterise the existence of certain (weakly) compact multipliers of the second dual of symmetric abstract Segal algebras in both the group algebra $L^{1}(G)$ and the Fourier algebra $A(G)$ of a locally compact group G.

We show that there is no nontrivial idempotent in the reduced group $\ell ^p$-operator algebra $B^p_r(F_n)$ of the free group $F_n$ on n generators for each positive integer n.

Given a minimal action $\alpha $ of a countable group on the Cantor set, we show that the alternating full group $\mathsf {A}(\alpha )$ is non-amenable if and only if the topological full group $\mathsf {F}(\alpha )$ is $C^*$ -simple. This implies, for instance, that the Elek–Monod example of non-amenable topological full group coming from a Cantor minimal $\mathbb {Z}^2$ -system is $C^*$ -simple.

Let X be a real prehomogeneous vector space under a reductive group G, such that X is an absolutely spherical G-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz–Bruhat functions on X against generalized matrix coefficients of admissible representations of $G(\mathbb {R})$ , twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation, as well as a functional equation in terms of abstract $\gamma $ -factors. This subsumes the archimedean zeta integrals of Godement–Jacquet, those of Sato–Shintani (in the spherical case), and the previous works of Bopp–Rubenthaler. The proof of functional equations is based on Knop’s results on Capelli operators.

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

We construct measures whose Fourier coefficients exhibit extreme behaviour among those whose support have Hausdorff h-measure zero.

This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions.

More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions.

These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

Let G be a complex semisimple Lie group and H a complex closed connected subgroup. Let and be their Lie algebras. We prove that the regular representation of G in $L^2(G/H)$ is tempered if and only if the orthogonal of in contains regular elements by showing simultaneously the equivalence to other striking conditions, such as has a solvable limit algebra.

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math. 397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys. 39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math. 308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of $\mathbb {R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb {R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order $p^{m}q^{n}$ , when one of the exponents is $\leq 6$ or when $p^{m-2}<q^{4}$ , and also prove that a tiling subset of a cyclic group of order $p_{1}^{m}p_{2}\dotsm p_{n}$ is spectral.

We show that the universal minimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation-invariant Boolean algebras of subsets of the group satisfying a higher-order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.

A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B.

In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

We show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$ .

This thesis is at the intersection of dynamics, probability and model theory. It focuses on a specialization of the notion of amenability: unique ergodicity.

Let G be a Polish group, i.e., a topological group whose topology is separable and completely metrizable. We call a G-flow the action of G on a compact space. A G-flow is said to be minimal if every orbit is dense.

A famous theorem of Ellis states that any Polish group G admits a unique universal minimal flow that we denote ${\mathrm {M}}(G)$ . This means that for any minimal G-flow X there is a surjective G-map from ${\mathrm {M}}(G)$ to X. G is said to be amenable if every G-flow admits an invariant probability measure, and uniquely ergodic if every minimal flow admits a unique invariant probability measure.

The notion of unique ergodicity relating to a group was introduced by Angel, Kechris and Lyons. They also ask the following question which is the main focus of the thesis: Let G be an amenable Polish group with metrizable universal minimal flow, is G uniquely ergodic?

Note that unique ergodicity is an interesting notion only for relatively large groups, as it is proved in the last chapter of this thesis that locally compact non compact Polish groups are never uniquely ergodic. This result is joint work with Andy Zucker.

The thesis includes proofs of unique ergodicity of groups with interesting universal minimal flows, namely the automorphism group of the semigeneric directed graph and the automorphism group of the $2$ -graph.

It also includes a theorem stating that under some hypothesis on a $\omega $ -categorical structure M, the logic action of ${\mathrm {Aut}}(M)$ on ${\mathrm {LO}}(M)$ , the compact space of linear orders on M, is uniquely ergodic. This implies unique ergodicity for the group if its universal minimal flow happens to be the space of linear orderings. It can also be used to prove non-amenability of some groups for which the action of ${\mathrm {Aut}}(M)$ on ${\mathrm {LO}}(M)$ is not minimal. This result is joint work with Todor Tsankov.

Finally, the thesis also presents a proof that under the assumption that the universal minimal flows involved are metrizable, unique ergodicity is stable under group extensions. This result is joint work with Andy Zucker.

Abstract prepared by Colin Jahel.

E-mail: cjahel@andrew.cmu.edu

URL: http://math.univ-lyon1.fr/~jahel/doc/These.pdf

We investigate the notion of relatively amenable topological action and show that the action of Thompson’s group T on $S^1$ is relatively amenable with respect to Thompson’s group F. We use this to conclude that F is exact if and only if T is exact. Moreover, we prove that the groupoid of germs of the action of T on $S^1$ is Borel amenable.