Patterns in sets of fractional dimension and harmonic analysis in non-linear dynamics
- Tuomas Sahlsten (Aalto University, Finland)
- Wednesday 15 May 2022, 15:00
- Room 310, Watson Building
It goes to the classical works of Szemerédi et al. to study the size of subsets of integers avoiding linear or non-linear progressions. Analogous problem of this in the real line goes back to the works of Bourgain and several subsequent papers, where for sets of zero Lebesgue measure one typically imposes some Fourier analytic assumptions on the set (e.g. the existence of a Rajchman measure whose Fourier decay rates relates closely to the Frostman properties of the measure). In fact, the works of Keleti and Shmerkin show that some assumptions are necessary always in this regime to find arithmetic progressions, but it is not clear what is the exact assumption needed from the set to find certain patterns. The examples here are characteristic to linear patterns, so it is plausible for non-linear patterns one might not need any assumptions. Recently, in a joint work with B. Kuca and T. Orponen, we considered a non-linear Sárközy-type problem in the plane, where we did not need any assumptions. Instead, we exploited the non-linearity of the pattern by connecting it to a kind of spectral gap property of "parabolic minimeasures". This still leaves open the question of finding linear progressions. Is there some more geometric assumption needed to always find arithmetic progressions in a set of fractional dimension? In a recent joint work with C. Stevens we studied this problem in the specific case of attractors of C2 dynamical systems, and connected the Fourier analytic assumptions to suitable non-linearity of the dynamics. In this talk we will attempt to give a very gentle and general introduction to this field and the methods we use.
Random substitutions, Rauzy fractals and the Pisot conjecture
- Dan Rust (The Open University, UK)
- Monday 16 May 2022, 14:10
- Room 310, Watson Building
A substitution is an iterative rule for replacing letters with words, like a↦ab, b↦a, the Fibonacci substitution. The Pisot property means that all of the eigenvalues of the associated incidence matrix lie in the open unit disc, except for the leading eigenvalue, which is real and greater than 1. The Pisot conjecture can be formulated in terms of a related fractal called the Rauzy fractal, whose generating IFS is 'dual' to the substitution. In this talk, we will introduce these concepts and explain how a new approach that introduces randomness may help to circumvent some of the biggest obstacles usually encountered when tackling the Pisot conjecture.
This is based on joint work with P. Gohlke, A. Mitchell and T. Samuel.
Random Lochs Theorem
- Charlene Kalle (Leiden University, The Netherlands)
- Monday 16 May 2022, 13:10
- Room 310, Watson Building
Lochs' theorem from 1964 concerns the number of continued fraction digits of a real number one can determine from knowing the first n decimal digits of this number. In 1999 Bosma, Dajani and Kraaikamp generalised this result to other number systems and placed it in a dynamical framework, where the statement relates the sizes of the cylinder sets of the systems involved. In this talk we will extend Lochs’ theorem to random dynamical systems and provide various examples of random number systems to which it applies.
This is based on joint work with E. Verbitskiy and B. Zeegers.
Measures arising from regular sequences: existence, ergodicity, and spectral classification
- Neil Manibo (Open University, UK)
- Thursday 5 May 2022, 15:00
- Room R17/18, Watson Building
In this talk, we will discuss how to build probability measures from regular sequences, which are generalisations of automatic sequences which are not restricted to be bounded. We provide sufficient conditions which guarantee the existence of the measures and their ergodicity under a suitable subgroup of the torus. We also provide a complete classification of their Lebesgue spectral type (pure point, absolutely continuous, singular continuous) dependent on quantities derivable from the linear representation of the sequence (joint spectral radius and Lyapunov exponents).
This is based on joint works with M. Coons, J. Evans, and Z. Groth.
Index theory and boundary value problems for general first-order elliptic differential operators
- Lashi Bandara (Brunel University London, UK)
- Thursday 31 March 2022, 15:00
- Room WG12, Aston Webb
Connections between index theory and boundary value problems are an old topic, dating back to the seminal work of Atiyah-Patodi-Singer in the mid-70s where they proved the famed APS Index Theorem for Dirac-type operators. From relative index theory arising in the study of positive scalar curvature metrics to a rigorous understanding of the chiral anomaly for the electron in particle physics, this index theorem has been a central tool to many aspects of modern mathematics.
APS showed that local boundary conditions are topologically obstructed for index theory. Therefore, a central theme emerging from the work of APS is the significance of non-local boundary conditions for first-order elliptic differential operators. An important contribution from APS was to demonstrate how their crucial non-local boundary condition for the index theorem could be obtained by a spectral projection associated to a so-called adapted boundary operator. In their application, this was a self-adjoint first-order elliptic differential operator.
The work of APS generated tremendous amount of activity in the topic from the mid-70s onwards, culminating with the Bär-Ballmann framework in 2010. This is a comprehensive machine useful to study elliptic boundary value problems for first-order elliptic operators on measured manifolds with compact and smooth boundary. It also featured an alternative and conceptual reformulation of the famous relative index theorem from the point of view of boundary value problems. However, as with other generalisations, a fundamental assumption in their work was that an adapted boundary operator can always be chosen self-adjoint.
Many operators, including all Dirac-type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. This operator has physical significance, arising in the study of the delta baryon, analogous to the way in which the Atiyah-Singer Dirac operator arises in the study of the electron. However, not only does the Rarita-Schwinger operator fail to be of Dirac-type, it can be shown that outside of trivial geometric situations, this operator can never admit a self-adjoint adapted boundary operator.
In this talk, I will present work with Bär where we extend the theory for first-order elliptic differential operators to full generality. That is, we make no assumptions on the spectral theory of the adapted boundary operator. The ellipticity of the original operator allows us to show that, modulo a lower order additive perturbation, the adapted boundary operator is in fact bi-sectorial. Identifying the spectral theory makes the problem tractable, although departure from self-adjointness significantly complicates the analysis. Therefore, we employ a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory, and maximal regularity to extend the Bär-Ballman framework to the fully general situation.
Time permitting, I will also talk about recent work on the relative index theorem for general first-order elliptic differential operators, possible harmonic-analytic perspectives of the APS index theorem, as well as recent developments in the study of noncompact boundary, Lipschitz boundary, and problems in Lp.
Dimensions of random cookie-cutter-like sets
- Wafa Ben Saad (Universität Bremen, Germany)
- Thursday 10 March 2022, 15:00
The random cookie-cutter-like sets are defined as the limit sets of a sequence of random cookie cutter mappings. By introducing the weak Gibbs-like measures, we study the fractal dimensions of these random sets and show that the Hausdorff dimension, the packing dimension and the box-counting dimension coincide and are almost surely equal to the unique zero of the topological pressure function.
On some recents results for the Lorentz gas with infinite horizon
- Dalia Terhesiu (Universiteit Leiden, The Netherlands)
- Thursday 03 March 2022, 15:00
In the first part of the talk I will recall recent local limit results including the very recent result with Melbourne and Pene on local large deviation. In the second part, I recall the set up of Lorentz gases with scatterer size going to zero. In joint work with Balint and Bruin, we obtain a precise understanding of the allowed path of taking limits as the size of the scatterers goes to zero and time goes to infinity so that the Central Limit Law (with non-standard normalization) persists. This type of joint law is, at some extent, in the gist of Boltzmann–Grad limit. Further, we obtain joint local limit theorems and mixing as the size of the scatterers goes to zero and time goes to infinity and I will comment on these.
Fourier transforms and nonlinear dynamics
- Connor Stevens (University of Manchester, UK)
- Thursday 10 February 2022, 15:00
Currently there is a great deal of research being produced in the area of Fourier transforms for measures. There is particular interest in when such objects exhibit (polynomial) decay. Much of this research in the context of nonlinear-map invariant measures stems from the ground-breaking paper of Bourgain–Dyatlov (2017) on Patterson–Sullivan measures for Hyperbolic Surfaces. In this talk we will discuss some of the methods used in this paper, and how large deviation theory can be used to apply their main tool of Discretized Sum-Product theory to the setting of expanding nonlinear maps rather than Fuchsian groups. We will conclude by looking at the assumptions necessary to use the Sum-Product theory, and heuristically how to prove these assumptions.
Notions of disorder for random substitutions
- Philipp Gohlke (Universität Bielefeld, Germany)
- Thursday 09 December 2021, 15:00
Substitutions are a classic tool to produce self-similar structures with a long-range order. The corresponding shift-dynamical systems can be deemed to be relatively ordered as they have vanishing entropy, and are often isomorphic to a rotation on a locally compact abelian group. By locally randomizing the substitution rule, we obtain dynamical systems that are much more disordered in a measure-theoretic, topological and combinatorial sense. At the same time, under appropriate conditions, they maintain long-range correlations, presenting themselves in a non-trivial pure point part of the diffraction measure. In this talk, we will discuss how to assess several notions of disorder in the context of random substitutions, including quantitative bounds for topological and measure-theoretic entropy.
This talk is based on joint work with A. Mitchell, D. Rust, T. Samuel, and T. Spindeler.
Complexity results for beta-expansions
- Simon Baker (University of Birmingham, UK)
- Thursday 25 November 2021, 15:00
β-expansions are a simple generalisation of the well known binary/ternary/decimal representations of real numbers. One of the most interesting properties they have is that a real number x almost surely has uncountably many beta-expansions. This means one can ask interesting questions about the size and complexity of the set of expansions. In this talk I will survey some results on the complexity of this set. The goal of the talk is to give an introduction to this topic and to the techniques involved. I will also pose some open questions.
Geometric functionals of fractal percolation
- Steffen Winter (KIT, Germany)
- Thursday 18 November 2021, 15:00
Fractal percolation is a family of random self-similar sets suggested by Mandelbrot in the seventies to model certain aspects of turbulence. It exhibits a dramatic topological phase transition, changing abruptly from a dust-like structure to the appearance of a system spanning cluster. The transition points are unknown and difficult to estimate, and beyond the fractal dimension not so much is known about the geometry of these sets. It is a natural question whether geometric functionals such as intrinsic volumes can provide further insights.
We study some geometric functionals of the fractal percolation process F, which arise as suitably rescaled limits of intrinsic volumes of finite approximations of F. We establish the almost sure existence of these limit functionals, clarify their structure and obtain explicit formulas for their expectations and variances as well as for their finite approximations. The approach is similar to fractal curvatures but in contrast the new functionals can be determined explicitly and approximated well from simulations. Joint work with M. Klatt.
Path-dependent shrinking target problem in generic affine iterated function systems
- Lingmin Liao (LAMA, UPEC, France)
- Thursday 11 November 2021, 10:00
The shrinking target problem studies the Hausdorff dimension of the set of points in a metric space whose orbits under a transformation hit a family of shrinking balls infinitely often. We consider a variation of such a problem by allowing the radii of the shrinking balls to depend on the path of the point itself in affine iterated function systems. It turns out that generically the Hausdorff dimension of this path-dependent shrinking target set is given by the zero point of a certain limsup pressure function. This is a joint work with H. Koivusalo and M. Rams.
Computational and dynamic complexity in shift spaces
- Robert Leek (University of Birmingham, UK)
- Thursday 28 October 2021, 15:00
We will discuss some results from the article The relationship between word complexity and computational complexity in subshifts by R. Pavlov and P. Vanier, as well as related papers. We will not assume any background on Turing reducibilty nor degrees.
Toral Anosov diffeomorphisms with computable resonances
- Julia Slipantschuk (University of Warwick, UK)
- Thursday 21 October 2021, 15:00
In the one-dimensional setting, Blaschke products give rise to analytic expanding circle maps for which the entire spectrum of the (compact) transfer operator is computable. Inspired by these examples, in this talk we will present a class of Anosov diffeomorphisms on the torus, constructed using Blaschke factors, for which the spectrum of transfer operators defined on a suitable anisotropic Hilbert spaces can be determined explicitly and related to the dynamical features of the underlying maps