# Ergodic Theory and Dynamical Systems Seminar

*The seminars are held on Tuesdays at 14:00 in B3.02*.

## Term 3

**Organizers: **Steve Cantrell, Joel Moreira, Han Yu

**April 23, 2024****Andreas Mountakis (University of Warwick)***Title: Finding product sets in some classes of amenable groups*

Abstract: In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erdős about sumsets in large subsets of the natural numbers. In this talk, we will discuss an extension of the result of the previous authors to several important classes of amenable groups, including finitely generated virtually nilpotent groups, and all abelian groups (G,+) with the property that the subgroup 2G has finite index. This is joint work with Dimitrios Charamaras.**April 30, 2024****Tom Kempton (University of Manchester)***Title:*The dynamics of the Fibonacci Partition FunctionAbstract: The Fibonacci partition function R(n) counts the number of ways of representing a natural number n as the sum of distinct Fibonacci numbers. For example, R(6)=2 since 6=5+1 and 6=3+2+1. An explicit formula for R(n) was recently given by Chow and Slattery. In this talk we express R(n) interms of ergodic sums over an irrational rotation, which allows us to prove lots of statements about the local structure of R(n).

**May 7, 2024****Sven Sandfeldt (KTH)***Title: Centralizer rigidity for partially hyperbolic diffeomorphisms on nilmanifolds*

Abstract: The smooth centralizer of a diffeomorphism f is the set of smooth maps g that commute with f. When f is an Anosov diffeomorphism on a nilmanifold, then a result by Rodriguez Hertz and Wang shows that: If the centralizer of f is higher rank, then f is smoothly conjugated to an affine map. In particular, for Anosov diffeomorphisms f, the only mechanism giving f a large centralizer is a smooth conjugacy to an algebraic model. In this talk, I will speak about rigidity of partially hyperbolic diffeomorphisms on nilmanifolds with large centralizer, extending the case for Anosov diffeomorphisms.

**May 14, 2024****Ian Morris (QMUL)***Title: Exceptional projections of self-affine sets.*

Abstract: If a subset X of R^{d}is projected onto a linear subspace then the Hausdorff dimension of its image is trivially bounded above by the rank of the projection and by the dimension of the set X itself. When the Hausdorff dimension of the image is smaller than both of these values the projection is called an*exceptional*projection for the set X. By the classical theorem of Marstrand, the set of exceptional projections of a Borel set always has Lebegue measure zero when considered as a subset of the relevant Grassmannian. I will describe some results from an ongoing systematic study of the exceptional projections of self-affine sets, including an example of a strongly irreducible self-affine set whose set of exceptional projections includes a nontrivial subvariety of the Grassmannian. This is joint work with Çağrı Sert.**May 21, 2024****Alexander Baumgartner (Warwick)***Title: Complex Continued Fractions and Hyperbolic Geometry*

Abstract: There is a deep link between continued fractions and the statistical behavior of geodesics on the modular surface. In joint work with Mark Pollicott, we studied the relationship between higher dimensional analogues of these notions. In particular, we study the link between complex-valued continued fractions and the dynamical properties of certain three-dimensional non-compact manifolds. We show that the maximal distance a geodesic travels away from its starting point obeys a Fréchet Distribution for large timeframes.**May 28, 2024****Tom Rush (Warwick)***Title: Thermodynamic formalism for random matrix products*

Abstract: Given a finite collection of GL(𝑑,𝑅) matrices, one can define a natural pressure function. In this talk, I will discuss results concerning equilibrium states and analyticity of the pressure function, as well as some of its applications to multifractal analysis**June 4, 2024****Pouya Honarya (University of Zurich)***Talk cancelled. New talk given by***Zhiyuan Zang (Imperial)***Title: Exponential mixing of 3D Anosov flows*

**June 11, 2024****Simon Baker (Loughborough University)**

*Title:*Fourier decay via L^2 flatteningAbstract: In this talk I will discuss a new approach for bounding the Fourier transform of a dynamically defined measure. Examples of measures to which this approach can be applied include self-similar measures, Patterson-Sullivan measures, Gibbs measures for conformal IFSs, and stationary measures for certain non-conformal IFSs. I will focus on the simplest case of self-similar measures. I will also explain how these Fourier transform bounds can be used to derive new equidistribution results. This talk is based on a joint work with Osama Khalil and Tuomas Sahlsten.**June 18, 2024****No seminar this week.****June 25, 2024****Speaker TBC***Title: TBA*

Abstract: TBA

## Term 2

**Organizers: **Cagri Sert, Julia Slipantschuk, Han Yu

**January 9, 2024****Stephen Cantrell (University of Warwick)***Title: Sparse spectrally rigid sets for negatively curved manifolds*

Abstract: Suppose that (M,g) is a closed, negatively curved Riemannian manifold. Given another such manifold (M',g') how do we determine whether (M,g) and (M',g') are the same (isometric)? It is conjectured that the isometry class of (M,g) is determined by its marked length spectrum: the lengths of closed geodesics on (M,g) ordered in a natural way. In this talk we discuss the following question. Is there a (hopefully `small') set of closed geodesics that determines the full marked length spectrum of (M,g)? That is, is there a small set of closed geodesics such that if (M,g) and (M',g') assign the same lengths to these geodesics then they must assign the same lengths to all closed geodesics (i.e. have the same marked length spectrum)?**January 16, 2024****Thomas Ward***Title: Hidden dynamical zeta functions*

Abstract: I will discuss examples of a phenomenon first observed (to my knowledge) by Yash Puri. Certain - as it turns out, many - sequences that arise naturally in combinatorics or number theory fail to count periodic points (that is, be the coefficients of a dynamical zeta function) but succeed after some modest modification. Examples include the Stirling numbers and linear recurrence sequences.**January 23, 2024****Sophie Grivaux (CNRS, Lille)***Title: Some new results regarding convergence under $\times_q$ of*

$\times_p$ invariant measures on the circle

Abstract: For each integer \(n\ge 1\), denote by \(T_{n}\) the map \(x\mapsto nx\mod 1\) from the circle group \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we will show that generically, a continuous \(T_{p}\)-invariant probability measure $\mu$ on \(\mathbb{T}\) is such that \((T_{q^{n}}\mu )_{n\ge 0}\) does not converge \(w^{*}\) to the Lebesgue measure on $\mathbb{T}$. This disproves Conjecture (C3) from a 1988 paper by R. Lyons, which is a stronger version of Furstenberg's rigidity conjecture on $\times_p$ and $\times_q$ invariant measures on $\mathbb{T}$, and complements previous results by Johnson and Rudolph. If time permits, I will also present some generalizations of this result concerning convergence to the Lebesgue measure of sequences of the form \((T_{c_{n}}\mu )_{n\ge 0}\), as well as some extensions to the multidimensional setting. The talk will be based on a joint work with Catalin Badea (Lille). **January 30, 2024****Robert Simon (London School of Economics)***Title: Paradoxical Decompositions and Colouring Rules*

Abstract: A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measure preserving tranformations of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We demonstrate paradoxical colouring rules defined via u.s.c. convex valued correspondences (if the colours b and c are acceptable by the rule than so are all convex combinations of b and c). This connects measure theoretic paradoxes to problems of optimisation and shows that there is a continuous mapping from bounded group-invariant measurable functions to itself that doesn't have a fixed point (but does has a fixed point in non measurable functions).**February 6, 2024****Kasun Fernando (Brunel University, London)***Title: The Bootstrap for Dynamical Systems*

Abstract: A dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using statistical methods. When measuring the uncertainty of such parameter estimations, the bootstrap stands out as a simple but powerful technique. In this talk, I will discuss the bootstrap for the Birkhoff averages of expanding maps and establish not only its consistency but also its second-order accuracy using the*continuous*first-order Edgeworth expansion.**February 13, 2024****Jerome Buzzi (CNRS, Orsay)***Title: Strong Positive Recurrence for diffeomorphisms*

Abstract:We introduce a new form of nonuniform hyperbolicity, which we call Strong Positive Recurrence (SPR) after a similar property studied in the symbolic setting by Gurevic and Sarig (themselves inspired by Vere-Jones’s geometric ergodicity property of Markov chains).This property implies a spectral gap property with many consequences: exponential mixing up to a period, limit theorems,…We shall explain that SPR is equivalent to a certain continuity property of Lyapunov exponents which is satisfied by all smooth surface diffeomorphisms with positive entropy but also holds for many higher dimensional dynamics.Joint work with Sylvain CROVISIER and Omri SARIG.**February 20, 2024****Carlangelo Liverani (Roma Tor Vergata)***Title: Globally coupled maps: Statistical properties and phase transitions*

Abstract: I will discuss infinite systems of globally coupled maps. The goal is to develop a general bifurcation theory that describes what happens when the coupling strength varies.

As an application, we show that phase transitions can occur naturally in a system of globally coupled Anosov maps. (Work in collaboration with Wael Bahsoun)**February 27, 2024****Martin Leguil (Ecole Polytechnique)***Title: Rigidity of u-Gibbs measures for certain Anosov diffeomorphisms of the 3-torus*

Abstract: we consider Anosov diffeomorphisms of the 3-torus $\mathbb{T}^3$ which admit a partially hyperbolic splitting $\mathbb{T}^3 = E^s \oplus E^c \oplus E^u$ whose central direction $E^c$ is uniformly expanded. We may consider the 2-dimensional unstable foliation $W^{cu}$ tangent to $E^c \oplus E^u$, but also the 1-dimensional strong unstable foliation $W^u$ tangent to $E^u$. The behavior of $W^{cu}$ is reasonably well understood; in particular, such systems have a unique invariant measure whose disintegrations along the leaves of $W^{cu}$ are absolutely continuous: the*SRB measure*. The behavior of $W^u$ is less understood; we can similarly consider the class of measures whose disintegrations along the leaves of $W^u$ are absolutely continuous, the so-called*u-Gibbs measures*. It is well-known that the SRB measure is u-Gibbs; conversely, in a joint work with S. Alvarez, D. Obata and B. Santiago, we show that in a neighborhood of conservative systems, if the strong bundles $E^s$ and $E^u$ are not jointly integrable, then there exists a unique u-Gibbs measure, which is the SRB measure.**March 5, 2024****Serge Troubetzkoy (Aix-Marseille University)***Title: Interval maps with dense periodicity*Abstract: I will report on joint work with J. Bobok, J. Činč, and P. Oprocha. We study the set $CP$ of interval maps with dense set of periodic points and its closure $\overline{CP}$ equipped with the metric of uniform convergence. I will describe some topological properties of theses spaces, give a dynamical characterization of $\overline{CP}$, describe some dynamical properties which hold generically in these spaces hold, and discuss the structure of conjugacy classes of such maps by homeomorphisms.**March 12, 2024****John Smillie (University of Warwick)***Title: Polygonal billiards and parabolic flows*

Abstract: I want to start with a simple question about billiards and show how this problem motivatesthe idea of rescaling the time parameter as a tool in parabolic dynamics. I will then explain how these ideas suggest an approach to some more complicated mathematical questions about polygonal billiards.

## Term 1

**Organizers: **Joel Moreira, Julia Slipantschuk, Timothée Bénard

**September 12, 2023****Anthony Quas (University of Victoria)***Title: Lyapunov Exponents and Noise*

Abstract: TBA-
**October 3, 2023****Yves Benoist (Universit***é*Paris-Saclay)*Title: Convolution and square on abelian groups.*

Abstract: The aim of this talk will be to construct functions on a cyclic group of odd order whose ''convolution square'' is proportional to their square. For that, we will have to interpret the cyclic group as a subgroup of an abelian variety with complex multiplication, and to use the modularity properties of their theta functions. -
**October 10, 2023****Konstantinos Tsinas***Title: Ergodic averages along primes*

Abstract: We study the limiting behavior of multiple ergodic averages along sequences evaluated at primes. Building on the result of Frantzikinakis, Host, and Kra, who established (in the most general setting known) the corresponding convergence theorem in the case that the sequences are integer polynomials, we generalize their result to other sequences of polynomial growth. The most prominent examples in our work are the fractional powers $\lfloor{n^c}\rfloor$, where $c$ is a positive non-integer. We prove that sets of positive density contain arbitrarily long arithmetic progressions with common difference $\lfloor{ p^c} \rfloor$, where $p$ denotes a prime, along with a few more mean convergence theorems and equidistribution results in nilmanifolds. Our methods rely on a recent deep theorem of Matom\"{a}ki, Shao, Tao, and Ter\"{a}v\"{a}inen on the Gowers uniformity of the von Mangoldt function in short intervals, an approximation of our functions with polynomials with good equidistribution properties and a lifting trick that allows someone to pass from ${\mathbb Z}$-actions on a probability space to ${\mathbb R}$-actions. **October 17, 2023****Roberto Castorrini***Title: Transfer operators, spectral gap and thermodynamic formalism: from smooth to discontinuous dynamical systems*Abstract: I will provide a brief overview of the functional approach used to analyze the statistical properties of a dynamical system, focusing on its main objective: determining a suitable Banach space that minimizes the 'non-compact' (essential) part of the spectrum of the associated transfer operator. Optimal outcomes regarding the essential spectrum for smooth hyperbolic dynamical systems are attained by employing thermodynamic formalism techniques, which utilize a variational expression for subadditive topological pressure. Drawing from a recent joint work with V. Baladi, I will illustrate similar results for systems with discontinuities, particularly piecewise expanding maps in finite dimensions.**October 24, 2023****Irving Calderón***Title: Explicit spectral gap for Schottky subgroups of $\mathrm{SL} (2, \mathbb{Z})$*

Abstract: Let $\mathcal{F}$ be a family of finite coverings of a hyperbolic surface $S$. A spectral gap of $\mathcal{F}$ is an interval $I = [0, \varepsilon]$ such that the eigenvalues in $I$ (counted with multiplicity) of the Laplacian $\Delta_S$ of $S$ and $\Delta_X$, any $X ∈ \mathcal{F}$, are the same. I will present a joint work with M. Magee where we give a spectral gap for congruence coverings when $S$ is the surface associated to a Schottky subgroup of $\mathrm{SL} (2, \mathbb{Z})$ with thick enough limit set. The proof exploits the link between eigenvalues of the Laplacian and zeros of dynamical zeta functions attached to $S$ via the thermodynamic formalism.-
**October 31, 2023****François Ledrappier***Title: Dimension of limit sets for Anosov representations*

Abstract: We consider the action of discrete finitely generated subgroup of matrices on the space of flags. Under hyperbolicity and non-degeneracy conditions, we can estimate the dimension of minimal invariant sets. The proofs use properties of random walks on the group. This is joint work with Pablo Lessa (Montevideo). -
**November 7, 2023****Amlan Banaji (Loughborough University)***Title: Fourier decay of fractal measures and their pushforwards*

Abstract: Determining when the Fourier transform of a measure decays to zero as a function of the frequency, and estimating the speed of decay if so, is an important problem. We will discuss this problem in relation to fractal measures arising from iterated function systems (IFSs), explaining that whilst in the strictly self-similar setting this depends in a delicate way on the number-theoretic properties of the IFS, systems with non-linearity often result in good decay. In particular, in joint work with Simon Baker, we use a disintegration technique to prove that the Fourier transform of non-linear pushforwards of a very general class of fractal measures decay at a polynomial rate. Combining this with a recent result of Algom – Rodriguez Hertz – Wang and Baker – Sahlsten, we prove that for any IFS on the line consisting of analytic contractions, at least one of which is not affine, every non-atomic self-conformal measure exhibits polynomial Fourier decay. This has interesting applications, for example related to the presence of normal numbers in fractal sets. -
**November 14, 2023****Matteo Tanzi***Title: Uniformly Expanding Coupled Maps: Self-Consistent Transfer Operators and Propagation of Chaos*

Abstract: Recently, much progress has been made in the mathematical study of self-consistent transfer operators which describe the mean-field limit of globally coupled maps. Conditions for the existence of equilibrium measures (fixed points for the self-consistent transfer operator) have been given, and their stability under perturbations and linear response have been investigated. In this talk, I am going to describe some novel developments on dynamical systems made of N uniformly expanding coupled maps when N is finite but large. I will introduce self-consistent transfer operators that approximate the evolution of measures under the dynamics, and quantify this approximation explicitly with respect to N. Using this result, I will show that uniformly expanding coupled maps satisfy propagation of chaos when N tends to infinity, and I will characterize the absolutely continuous invariant measures for the finite dimensional system. -
**November 21, 2023****Li Dongchen***Title: Persistence of heterodimensional cycles*

Abstract: The existence of heterodimensional cycles is believed to be one of two basic mechanisms leading to non-hyperbolicity, where the other one is the existence of homoclinic tangencies. We show that any $C^r$ ($r=2,\ldots,\infty,\omega$) system having a heterodimensional cycle can be approximated in the $C^r$ topology by systems having robust heterodimensional cycles. This implies a counterpart for heterodimensional cycles to the well-known Newhouse theorem that every homoclinic tangency is C^r close to robust homoclinic tangencies. The result is based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-D\'iaz blenders. -
**December 5, 2023****Weikun He***Title: Dimension theory of groups of circle diffeomorphisms.*

Abstract: In this talk, we consider the action of a finitely generated group on the circle by analytic diffeomorphisms. We will discuss some results concerning the dimensions of objects arising from this action. More precisely, we will present connections among the dimension of minimal subsets, that of stationary measures, entropy of random walks, Lyapunov exponents and critical exponents. These can be viewed as generalizations of well-known results in the situation of PSL(2,R) acting on the circle.This talk is based on a joint work with Yuxiang Jiao and Disheng Xu. -
**December 12, 2023****Christian Wolf****Room B3.03***Title: Ergodic theory on coded shifts spaces*

Abstract:In this talk we present results about ergodic-theoretic properties ofcoded shift spaces. A coded shift space is defined as a closure of all bi-infiniteconcatenations of words from a fixed countable generating set. We derivesufficient conditions for the uniqueness of measures of maximal entropy andequilibrium states of H\"{o}lder continuous potentials based on the partition of the codedshift into its sequential set (sequences that are concatenations of generating words)and its residual set (sequences added under the closure). We also discussflexibility results for the entropy on the sequential and residual set. Finally, we presenta local structure theorem for intrinsically ergodic coded shift spaces which showsthat our results apply to a larger class of coded shift spaces compared to previous worksby Climenhaga, Climenhaga and Thompson, and Pavlov. The results presentedin this talk are joint work with Tamara Kucherenko and Martin Schmoll.

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Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

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