All talks will be held in the Schofield building room SCH 1.01.
Coffee breaks will take place in SCH.0.04 (on the ground floor opposite the main lecture theatre).
On Tuesday evening there will be an informal dinner at a nearby restaurant.
8th July
1-1:15 - Welcome
1:15-2:15 - Balazs Barany - A simultaneous dimension result via transversality
2:25-2:55 - George Bender - Points with finitely many base-q expansions
2:55-3:35 - Coffee break
3:35-4:35- Xiong Jin - A Chung-Fuchs type theorem for skew product dynamical systems
4:45-5:15 - Ana de Orellana - Fourier analysis for fractal measures
9th July
9:30-10:30 - Ian Morris - Linear images of self-affine sets
10:30-11:05 - Coffee break
11:05-11:35 - Tanja Schindler - Scaling properties of (generalised) Thue-Morse measures
11:45-12:15 - Dmytro Karvatskyi - Geometry of Cantorvals
12:15-1:45 - Lunch
1:45-2:15- Charlie Wilson - Some new results on hitting points infinitely often and limsup sets
2:25-3:25 - Leticia Pardo Simon - A transcendental map with a hair of Hausdorff dimension two
3:25-4:05 - Coffee break
4:05-4:35 - István Kolossváry - Assouad spectrum of Gatzouras-Lalley carpets
4:45-5:15 - William O'Regan - On the discretised ring theorem
6:15 - Dinner at the basin
10th July
9:00-10:00 - Karma Dajani - The matching phenomenon
10:10-10:40 - Sven van Golden - Infinitely generated self-affine sets and restricted digit sets for signed Lüroth expansions
10:40-11:15 - Coffee break
11:15-12:15 - Mark Pollicott - The dimension of the spectrum of Laplacian of the Sierpinski triangle
12:15 - Farewell
Titles and abstracts
Monday
1:15 - 2:15: Balázs Bárány (Budapest University of Technology and Economics)
Title: A simultaneous dimension result via transversality
Abstract: To study the dimension theory of general self-conformal IFSs with overlaps, a widely used method is to study parametrized families of such IFSs instead of individual ones. If such a family satisfies the so-called transversality condition, then it has been known for two decades that for every invariant ergodic measure defined on the symbolic space, there is a full-measure subset of parameters for which the Hausdorff dimension of the push-forward measure can be determined by the ratio entropy over Lyapunov exponent. In this talk, we show that we can choose a full-measure subset of parameters for which the assertion above holds for all ergodic measures simultaneously. We apply this result to the multifractal analysis of typical self-similar measures with overlaps. This is a joint work with Károly Simon and Adam Śpiewak.
2:25 - 2:55: George Bender (University of Birmingham)
Title: Points with finitely many base-q expansions
Abstract: We explore the methods used to prove the existence of points with exactly two base-q expansions and the extensions required to generalise results of this form.
3:35 - 4:35: Xiong Jin (University of Manchester)
Title: A Chung-Fuchs type theorem for skew product dynamical systems
Abstract: I will present a Chung-Fuchs type theorem for skew product dynamical systems such that for a measurable function on such a system, if its Birkhoff average converges to zero almost surely, and on typical fibres its Birkhoff sums have a non-trivial independent structure, then its associated generalised random walk oscillates, that is the supremum of the random walk equals to +infinity and the infimum equals to -infinity. I will also present some application of this theorem to Mandelbrot cascades acting on ergodic measures.
4:45 - 5:15 - Ana de Orellana (University of St Andrews)
Title: Fourier analysis for fractal measures
Abstract: The Fourier spectrum is a family of dimensions that, for measures, live between the Fourier and Sobolev dimensions, and is defined in terms of the Fourier transform of the measure. In this talk we will extend the well-known results of Kahane and Hare—Roginskaya to define the Fourier spectrum by considering only the Fourier coefficients of measures. Joint work with Marc Carnovale and Jonathan Fraser.
Tuesday
9:30 - 10:30 - Ian Morris (Queen Mary University of London )
Title: Linear images of self-affine sets
Abstract: In this talk I will describe how the thermodynamic formalism of self-affine sets can be extended to the context of arbitrary linear projections of self-affine sets. I will describe new estimates for the dimensions of such sets and new examples of self-affine sets which have large families of exceptional projections in the sense of Marstrand's theorem. This is joint work with Çağrı Sert.
11:05 - 11:35 - Tanja Schindler (Jagiellonian University)
Title: Scaling properties of (generalised) Thue-Morse measures
Abstract: The Thue-Morse measure and its generalisations are diffraction measures of simple aperiodic systems. Besides that, they are paradigmatic examples of purely singular continuous probability measures on the unit interval given as an infinite Riesz product. To study their scaling behaviour a classical method, the thermodynamic formalism can be used - which however has to be adapted to an unbounded potential. Besides seeing this method, we will also consider the L^q spectrum of those measures. This is joint work with M. Baake, P. Gohlke, and M. Kesseböhmer.
11:45 - 12:15 - Dmytro Karvatskyi (University of St Andrews)
Title: Geometry of Cantorvals
Abstract: We study the topological, metric and fractal properties of Cantorvals, a strange union of intervals and a fractal set. Cantorvals have been discovered independently a few times in various topological and dynamical contexts. For instance, Cantorval is one of three possible topological types of the set of subsums for a convergent positive series; it can also be a product of the arithmetical sums of two Cantor sets with zero Lebesgue measure; or it might be an attractor for an iterated function system.
1:45 - 2:15- Charlie Wilson (Exeter University)
Title: Some new results on hitting points infinitely often and limsup sets
Abstract: We examine some fairly easily stated measure theory problems about covering a space with sets and seeing which points are hit infinitely often. That is if we if fixing the measures of finite intersections, what can we say about the measure of the limsup. This will tie in to the Divergence Borel Cantelli Lemma and Erdos Chung inequality and show that there are limits to the use of such inequalities. We will go on to see that what we may feel is intuitively true and also what the current state of the art may indicate actually isn't- via some counterexamples that will be shown. The result, though a nice problem in isolation, does indeed have ramifications in many other areas of maths such as Probability theory, Diophantine approximation and Dynamical systems.
2:25 - 3:25 - Leticia Pardo Simon (University of Barcelona)
Title: A transcendental map with a hair of Hausdorff dimension two
Abstract: The Julia sets of many transcendental entire functions are structured as a collection of curves tending to infinity, called hairs. In all previously known examples, like for all exponential and cosine maps with this property, the union of all such curves has Hausdorff dimension two, while the dimension of each individual hair is one. In this talk I will explain how to construct an example of a Cantor bouquet Julia set with a hair of Hausdorff dimension two. This is based on work in progress with W. Cui, D. Martí-Pete and L. Rempe.
4:05-4:35 - István Kolossváry (Alfréd Rényi Institute of Mathematics)
Title: Assouad spectrum of Gatzouras-Lalley carpets
Abstract: The talk will focus on the fine local scaling properties of a class of self-affine fractal sets called Gatzouras-Lalley carpets by looking at its Assouad spectrum. This is a one parameter family of dimensions that interpolate between the box and (quasi-)Assouad dimension of the set. We will show a formula for the Assouad spectrum of all Gatzouras-Lalley carpets obtained by taking the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Time permitting, we will demonstrate a number of novel properties that our formula exhibits for dynamically invariant sets and give some hints of the proof strategy. Based on joint work with A. Banaji, J.M. Fraser and A. Rutar.
4:45- 5:15 - William O'Regan (University of Warwick)
Title: On the discretised ring theorem
Abstract: I will introduce and motivate the discrete version of a fractal and relate this to the usual notion of a fractal set. I will then survey the discretised ring theorem and related problems. Some of this talk will be based on joint work with Andras Mathe.
Wednesday
9:00-10:00 - Karma Dajani (University of Utrecht)
Title: The matching phenomenon
Abstract: Matching is a mysterious phenomenon which has recently been observed for certain parametrized interval maps in the deterministic and random settings. For example, α-continued fractions, α-symmetric binary maps as well as α-golden mean maps. Matching is the property that for each discontinuity point the orbits of the left and right limit merge after some finite number of steps and that the (expected value of the) derivatives of both orbits are also equal at that time; this assures the stability of this phenomenon under small perturbations of the parameter. Since most of the dynamical behaviour of systems are encoded in the possible trajectories of the discontinuity points, knowledge on when and how matching occurs can help in finding explicit expression for the natural invariant measure. Once such a measure is found, one is able to obtain essential information regarding the system, such as the frequency the orbits enter a specific region, the entropy, the Lyapunov exponents, mixing rates etc. I will illustrate this phenomenon with the parametrized family of symmetric doubling maps. This is joint work with C. Kalle.
10:10-10:40 - Sven van Golden (University of Birmingham)
Title: Infinitely generated self-affine sets and restricted digit sets for signed Lüroth expansions
Abstract: In recent years much work has been done towards finding the fractal dimensions of the limit sets of finite affine iterated function systems, also known as self-affine sets. Of particular interest are conditions under which the Hausdorff dimension of such sets equals the affinity dimension, a value introduced by Falconer in 1988. Even more recently, this concept has been extended to self-affine sets generated by infinite iterated function systems.
In this joint work with C. Kalle, S. Kombrink and T. Samuel we consider self-affine sets generated by countably infinite iterated function systems where each affine map has diagonal linear parts. We introduce a family of such self-affine sets in the plane that arise from the restricted digit sets for signed Lüroth expansions. Moreover, we study their vertical fibres to find conditions under which their Hausdorff dimensions equal the affinity dimension.
11:15-12:15 - Mark Pollicott (University of Warwick)
Title: The dimension of the spectrum of Laplacian of the Sierpinski triangle
Abstract: One can associate to the Sierpinski triangle (and related fractals) a laplacian operator, motivated by the study of the laplacian on riemannian manifolds. This is a subset of the real line whose (Hausdorff) dimension is an interesting additional numerical characteristic of the spectrum. We can estimate this numerical value. This is joint work with Julia Slipantschuk.