# Ergodic Theory and Fractal Days

Since 2013 we have been organising meetings on the topic of ergodic theory and fractals in Utrecht, Leiden and Delft at irregular time intervals. Details of the meetings can be found below. Besides the opportunity to hear lots of exciting talks, the purpose of these meetings is to encourage more regular contact between researchers in ergodic theory in the Netherlands. The title of the meetings should be interpreted broadly and we hope that the meetings will be of interest to people working in all areas of dynamical systems, ergodic theory and related topics.

The meetings are currently organised by Karma Dajani (k.dajani1@uu.nl), Robbert Fokkink (r.j.fokkink@tudelft.nl) and Charlene Kalle (kallecccj@math.leidenuniv.nl) and in the past also by Tom Kempton and Cor Kraaikamp. Please feel free to email one of the organisers if you have any questions.

The meeting in 2022 is supported by Delft University, Leiden University and Utrecht University. The meetings in 2013 and 2018 were supported by a seminar grant from STAR. The meetings in 2013 were also supported by an incidental support grant from NWO.

## 1 July 2022

Todays location will be announced later. Click here for directions. The schedule is as follows.

 10.00 - 10.45: TBA (TBA) Title: TBA 10.45 - 11.15: Coffee break 11.15 - 12.00: TBA (TBA) Title: TBA Abstract: TBA 12.00 - 14.00: Lunch 14.00 - 14.45: TBA (TBA) Title: TBA Abstract: click here. 14.45 - 15.15: Coffee break 15.15 - 16.00: TBA (TBA) Title: TBA Abstract: TBA

After the last talk we will go for drinks and dinner.

## 17 May 2019

On this day the mini-workshop Expansions and Substitutions was organised at Utrecht University.

## 1 June 2018

Todays location will be Room K in the EWI-building of the TU Delft. Click here for directions. The schedule is as follows.

 10.00 - 10.45: Liza Arzhakova (Leiden) Title: On the decimation of the Laurent polynomials Abstract: The algebraic Z^d - actions naturally appear in statistical mechanics as actions on the lattice. The Pontryagin duality implies a one-to-one correspondence between the algebraic Z^d - actions and modules over the ring of Laurent polynomials. Moreover, the decimation of the lattice corresponds in this context to the decimation of the polynomial, which is the central object of my research. I am interested in the asymptotic growth of the coefficients under the decimation procedure. In this talk I will prove an upper bound of the coefficient growth and show that the limit need not exist. 10.45 - 11.15: Coffee break 11.15 - 12.00: Kiko Kawamura (UNT) Title: Relationship between revolving sequences and self-similar sets Abstract: In 1987, Mizutani and Ito pointed out a close relationship between revolving sequences and Dragon, which is a famous tiling self-similar set, from the viewpoint of symbolic dynamical systems. We will show how their result can be generalized by a completely different approach. The talk will be presented with a lot of pictures; accessible even for undergraduate students. A few open problems will be introduced as well. 12.00 - 12.15: Break 12.15 - 13.00: Yuanyuan Yao (Shanghai) Title: On the structure of λ-Cantor set with overlaps Abstract: Let E_λ be the attractor of the iterated function system {x/3,(x+λ)/3,(x+2)/3} with λ ∈ (0,1). In 2004, Broomhead, Montaldi and Sidorov defined a finer family of self-similar set with overlap, which is totally self-similar. In this talk, we will give the necessary and sufficient condition for E_λ to be totally self-similar and will describe all the generating iterated function systems for E_λ when E_λ is totally self-similar. Besides, we discuss the properties of the spectrum of E_λ and give some examples where the spectrum can be explicitly determined. This is in joint work with Karma Dajani and Derong Kong. 13.00 - 14.30: Lunch break 14.30 - 15.15: Sara Munday (Pisa) Title: Pointwise convergence of Birkhoff averages for global observables Abstract: It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost everywhere zero. Furthermore, Aaronson has shown that a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit does not exist. In this talk, presenting joint work with Marco Lenci, I will outline a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions certain elements of $L^\infty$ are used, which will be generically called "global observables". This result applies to general systems but requires an hypothesis of "approximate partial averaging" on the observables. The idea behind the result, however, applies to more general situations, as will be shown with an example. Finally, time permitting, by means of counterexamples and numerical simulations, the question of finding the optimal class of observables for which a Birkhoff theorem holds for infinite-measure-preserving systems will be discussed. 15.15 - 15.45: Coffee break 15.45 - 16.30: Marta Maggioni (Leiden) Title: Invariant densities for random dynamical systems Abstract: We will describe invariant densities for any random system of piecewise linear maps that are expanding on average. More precisely, we provide a procedure to obtain an explicit formula for the density of an absolutely continuous invariant measure. This result generalises the method of Kopf (1990), valid in the deterministic setting. We conclude showing how this construction merges the results by Kempton (2014) and Suzuki (2017) on random β-transformations. Joint work with C. Kalle.

After the last talk we will go for drinks and dinner.

## 20 April 2018

Todays location will be Room 408 in the Snellius building of Leiden University, which is the building where the Mathematical Institute is located. Click here for directions. The schedule is as follows.

 10.00 - 10.45: Vasso Anagnostopoulou (Queen Mary) Title: Sturmian measures in ergodic optimisation Abstract: For a real-valued function f, an invariant probability measure is called f-maximising if it gives f a larger space average than any other invariant probability measure. Ergodic optimisation is the study of problems relating to maximising invariant measures and maximum ergodic averages. In this talk, we will discuss the role of a one-parameter family of measures, the Sturmian measures, in various problems in ergodic optimisation. 10.45 - 11.15: Coffee break 11.15 - 12.00: Ale Jan Homburg (Amsterdam) Title: On-off intermittency and chaotic walks Abstract: We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We discuss the appearance of on-off intermittency. A main ingredient is the equivalent description in terms of chaotic walks: random walks driven by the doubling map. 12.00 - 12.15: Break 12.15 - 13.00: Steven Berghout (Leiden) Title: On functions (factors) of Markov measures Abstract: Factors of Markov measures, or hidden Markov models, are usually not Markovian and possibly not even g-measures or Gibbs measures. Existence of a continuous measure disintegration is a sufficient condition for an analogous problem; when a factor of a g-measure is a g-measure. In this talk I will discuss the application of this condition to factors of Markov measures and its relation to existing results. 13.00 - 14.30: Lunch break 14.30 - 15.15: Wael Bahsoun (Loughborough) Title: Linear response for random dynamical systems Abstract: In this talk I will give a brief overview on the topic of linear response in dynamical systems. I will then discuss in details recent results on linear response for random compositions of maps and their applications to Gauss-Renyi maps and random Pomeau-Manneville maps . This is a joint work with Marks Ruziboev and Benoit Saussol. 15.15 - 15.45: Coffee break 15.45 - 16.30: Derong Kong (Leiden) Title: Numbers with simply normal β-expansions Abstract: When β ∈ (1,2) it is well-known that almost every x ∈ I_β =[0, 1/(β-1)] has a continuum of β-expansions. In this talk we show that there exists a critical base β_T ≈ 1.80194, the unique zero in (1,2] of the polynomial x^3-x^2-2x+1, such that for β ∈ (1, β_T] every interior point of I_β has a simply normal β-expansion, while for β ∈ (β_T, 2] there exists a point in I_β that does not have a simply normal β-expansion. This is a joint work with Simon Baker.

After the last talk we will go for drinks and dinner.

## 23 March 2018

Todays location will be Room 020 in the Buys Ballot building of Utrecht University. Click here for directions. The schedule is as follows.

 10.00 - 10.45: Tuomas Sahlsten (Manchester) Title: Additive combinatorics in ergodic theory Abstract: We will prove some new Fourier decay results on equilibrium states associated to sufficiently nonlinear Markov maps (such as fractal measures on badly approximable numbers and other examples). It turns out that after employing large deviation theory in thermodynamical formalism, the problem can be reformulated as a decay theorem for multiplicative convolutions, which in turn can be approached with methods from discretised sum-product theory in additive combinatorics. This work builds on the recent results of Jean Bourgain and Semyon Dyatlov (2017) on Fractal Uncertainty Principle and an earlier work with Thomas Jordan (2016). Joint work with Connor Stevens (Manchester) 10.45 - 11.15: Coffee break 11.15 - 12.00: Pieter Allaart (UNT) Title: Differentiability and Hölder spectra of a class of self-affine functions Abstract: In 1973, P. Lax published a surprising paper about the differentiability of Polya's space-filling curves. In 2006, H. Okamoto introduced a completely different one-parameter family of functions, whose differentiability structure is nonetheless quite similar to that of the Polya curve. In this talk I will explain that both functions are in fact members of the same general class. In addition to discussing differentiability of functions in this class, I will also show that their pointwise Hölder (or multifractal) spectrum is given by the multifractal formalism, though this does not seem to follow from standard multifractal theory. 12.00 - 12.15: Break 12.15 - 13.00: Niels Langeveld (Leiden) Title: Ito α-continued fractions, matching and holes Abstract: For Ito α-continued fractions we can write a condition in terms of a system with a hole such that, whenever the condition is satisfied, matching holds. This means that under forward iterations the orbit of α and α-1 coincide. From this condition we will explain interesting behavior around bad rationals. These are rationals that match but do not lie in a matching interval (an interval for which points match in a similar manner). Furthermore we will show that the set of α for which we do not have matching has full Hausdorff dimension. 13.00 - 14.30: Lunch break 14.30 - 15.15: Sabrina Kombrink (Lübeck) Title: Steiner formula for fractal sets Abstract: The famous Steiner formula for a non-empty compact convex subset K of d-dimensional Euclidean space states that the volume of the ε-parallel set of K can be expressed as a polynomial in ε of degree d. The coefficients of the polynomial carry important information on the geometry of the convex set, such as volume, surface area and Euler characteristic. For fractal sets the ε-parallel volume is more involved and cannot be written as an ordinary polynomial in ε. In this talk we discuss the behaviour of the ε-parallel volumes of certain fractals and analogues of the Steiner formula. Moreover we explore the geometric information which the analogues of the exponents and coefficients incorporate. 15.15 - 15.45: Coffee break 15.45 - 16.30: Jaap de Jonge (Delft) Title: Gaps in orbits of N-expansions Abstract: For the abstract, click here.

After the last talk we will go for drinks and dinner.

## 23 May 2014

Todays location will be Room 312 in the Snellius building of Leiden University, which is the building where the Mathematical Institute is located. Click here for directions. The schedule is as follows.

 11.30 - 12.30: Jonathan Fraser (Warwick) Title: Scaling scenery of (xm,xn) invariant measures Abstract: Using ergodic theory to study problems in geometry is not new, however, there have recently been some major advances in the fields of fractal geometry and geometric measure theory made by studying the dynamics of the process of 'zooming in' on fractal sets and measures. In particular, Hochman and Hochman-Shmerkin have recently developed ideas of Furstenberg to produce a rich and ripe theory. The dynamics of the blow-ups can be modelled using a 'CP-chain', which records both the point where we zoom-in, and the scenery which we then see. Thus far CP-chains have proved a powerful tool in studying geometric properties of self-similar measures, with applications to projection theorems and distance set problems. The aim of this talk is to motivate the study of CP-chains and attempt to extend the theory beyond the conformal setting. This will be done in the context of Bernoulli measures on self-affine Bedford-McMullen carpets, which are a first step towards a general study of (xm,xn) invariant measures on the 2-torus. This is based on joint work with Andrew Ferguson and Tuomas Sahlsten. 12.30 - 1.45: Lunch 1.45 - 2.45: Cor Kraaikamp (Delft) Title: Quilting Natural Extensions of Continued Fraction Expansions Abstract: In 1981, Hitoshi Nakada introduced a family of continued fraction maps, and studied their natural extensions. These are the Nakada alpha-expansions, which are defined for a parameter alpha between 0 and 1. These alpha-expansions played a key role in the revival of the interest in continued fraction expansions, and are up to today subject of thorough investigations. Using some extremely basic ideas called insertions' and singularizations' we will show that there is a strong relation between alpha-expansions for various values of alpha-expansions. In this talk I will show how far these ideas can be carried over, and how they can be used in other settings, e.g. for the so-called Rosen fractions.' 2.45 - 3.15: Coffee break 3.15 - 4.15: Carlo Carminati (Pisa) Title: Matching for a family of piecewise linear maps Abstract: We consider a 1-parameter family (Q_γ)_{γ ∈ ℝ} of piecewise linear maps, and we study how the metric entropy of Q_γ depends upon the parameter γ. Despite the simple nature of the system, the behaviour of the entropy is quite surprising: it is smooth outside a zero measure set (but not everywhere), and its graph displays a complicated self-similar structure. We shall show that this phenomenon is due to a special combinatorial feature, called matching property, which was first detected for the family of α-continued fractions. Indeed, we belive that understanding the mechanisms which rule the behaviour of the entropy for this family of piecewise maps might help to give an answer to some open questions about the family of α-continued fractions. (This is a work in progress with H. Bruin, S. Marmi and A. Profeti).

## 8 November 2013

Todays location will be Room 401 in the Snellius building of Leiden University, which is the building where the Mathematical Institute is located. Click here for directions. The schedule is as follows.

 11.30 - 12.30: Henk Bruin (Vienna) Title: Substitution shifts and renormalization for potentials Abstract: Thermodynamic formalism has been implemented in symbolic dynamics a long time ago. For example, Bowen showed that for a Holder potential V on a subshift of finite type, there are no phase transitions, which means that the pressure function β ↦ P(β V) := sup{ h(μ) + β ∫ V dμ} is analytic. Starting with Hofbauer, non-Holder potentials were studied and phase transitions detected. Hofbauer's examples relates directly to the Pomeau-Manneville map, which Baraviera-Leplaideur-Lopes related again to a particular substitution-based renormalization operator. In this talk, I want to report on joint work with Leplaideur how this scheme extends to non-trivial substitutions (Thue-Morse and Fibonacci), and discuss the importance and properties of the fixed points of the renormalization operator, their stable leaves, and time-permitting also the existence of phase transitions (in an attempt to build a model to explain the formation of one-dimensional quasi-crystals). 12.30 - 1.45: Lunch 1.45 - 2.45: Tomas Persson (Lund) Title: Limsup-sets of random covers Abstract: Suppose that we have a sequence of open subsets of a torus. We translate these open sets randomly and form the limsup-set, that is the set of points that are covered infinitely often by the translated open sets. I will talk about fractal properties that hold almost surely for such limsup-sets, and how such properties can be proved in a simple way using a certain lemma. This is based on this paper on arXiv. 2.45 - 3.15: Coffee break 3.15 - 4.15: Siamak Taati (Utrecht) Title: Some ergodic-theoretic problems in cellular automata Abstract: A cellular automaton is a topological dynamical system on a (multi-dimensional) shift space that commutes with the shift. In physics applications, the shift is seen as the spacial translation and the cellular automaton as the time evolution. Miyamoto (1979) and Lind (1984) proved that the iterates of the XOR cellular automaton (the Pascal triangle modulo 2) on a biased Bernoulli measure converges in density to the uniform Bernoulli measure. This is an analogue of a result of Johnson and Rudolph (1995) regarding the (2x, 3x) system, with 2x interpreted as the time dynamics and 3x as the space dynamics. Various extensions of the theorem of Miyamoto and Lind have been found, but they are all limited to cellular automata with algebraic structures. The phenomenon of convergence to a higher-entropy measure however seems to be far more general, with support from simulations and analogy with the second law of thermodynamics in nature. The limit measure is invariant under the cellular automaton, and therefore, a related question is to identify the invariant measures of a cellular automaton. Characterizing all the invariant measures of a cellular automaton could be difficult, but I will discuss a connection between the invariance of Gibbs measures (i.e., the measures used in statistical mechanics to describe thermodynamic equilibrium) and the presence of conservation laws in surjective cellular automata. As a corollary, we obtain a severe restriction on the invariant measures of two classes of surjective and reversible (= bijective) cellular automata with strong chaotic behavior. This is a joint work with Jarkko Kari.

After the last talk we will go for drinks and dinner.

## 13 September 2013

Todays location will be Room 204 (first talk) and Room 027 (last two talks) in the Minnaert building of Utrecht University. Click here for directions. The schedule is as follows.

 11.30 - 12.30: Valérie Berthé (Paris) Title: Symbolic discrepancy and continued fractions Abstract: We discuss in this lecture the notion of symbolic discrepancy that provides a measure of convergence toward letter densities in infinite words. We consider the particular case of words and shifts generated by continued fraction algorithms, by stressing the connections with corresponding Lyapunov exponents. Applications to spectral properties of associated symbolic dynamical systems will also be given under a generalization of the Pisot hypothesis. 12.30 - 1.45: Lunch 1.45 - 2.45: Evgeny Verbitskiy (Leiden) Title: Expansivity and homoclinic points in algebraic dynamics Abstract: After a short review of expansivity properties of algebraic dynamical systems and existence of homoclinic points in general, I will discuss a specific case of principal algebraic actions of the discrete Heisenberg group. Criterium for expansivity based on Allan's local principle will be presented, as well as the novel link to time-frequency analysis and difference equations with almost periodic coefficients. This is joint work with M. Göll and K. Schmidt. 2.45 - 3.15: Coffee break 3.15 - 4.15: Michel Dekking (Delft) Title: The isomorphism problem for substitution dynamical systems. Abstract: The first contribution to this problem was made 42 years ago by Coven and Keane, who solved the case of an alphabet of cardinality two. In 1989 Host and Parreau made a new contribution, but the general problem was still open. I will report on a recent solution by Coven, Keane and myself.

After the last talk we will go for drinks.

## 17 May 2013

Today's location will be Room 405 in the Snellius building of Leiden University, which is the building where the Mathematical Institute is located. Click here for directions. The schedule is as follows.

 11.30 - 12.30: Shigeki Akiyama (Tsukuba) Title: Height Reducing Problem Abstract: Given an algebraic number β. If there is a constant M such that each element of ℤ[β] is represented as a polynomial in β with integer coefficients not greater than M in modulus, we say that β has height reducing property (HRP). This problem aroze from constructions of self-affine tilings. I wish to give a brief review of this problem as well as some recent progress jointly doned with T.Zaimi. Interestingly, HRP is related to the effective version of Kronecker's approximation theorem. 12.30 - 1.45: Lunch 1.45 - 2.45: Karma Dajani (Utrecht) Title: Optimal Expansions in non-integer base Abstract: click here. 2.45 - 3.15: Coffee break 3.15 - 4.15: Nikita Sidorov (Manchester) Title: The doubling map with asymmetrical holes Abstract: Let T denote the doubling map on the interval [0,1] and let 0 < a < b < 1. Denote by J(a,b) the set of all x in [0,1] such that the T-orbit of x has an empty intersection with the interval (a,b). In my talk I am going to give a full description of (a,b) such that J(a,b) is of positive Hausdorff dimension. This description will involve balanced words and Sturmian sequences. I will also explain how this family of open dynamical systems provides explicit routes to chaos' previously observed numerically in conventional dynamical systems. This talk is based on a recent joint paper with Paul Glendinning.

After the last talk we will go for drinks and dinner.

## 4 March 2013

Today's location will be Room B2 on the ground floor of the Snellius building of Leiden University. This is where the Mathematical Institute is located. Click here for directions. The schedule is as follows.

 11.30 - 12.30: Mike Keane (Wesleyan) Title: The binomial transformation Abstract: TBA 12.30 - 1.45: Lunch 1.45 - 2.45: Pablo Shmerkin (Surrey) Title: Normal numbers and fractal measures Abstract: It is known from E. Borel that almost all real numbers are normal to all integer bases. On the other hand, it is conjectured that natural constants such as π, e or √2 are normal, but this problem is so far untractable. In the talk I will describe a new dynamical approach to an intermediate problem: are natural'' fractal measures supported on numbers normal to a given base? Our results are formulated in terms of an auxiliary flow that reflects the structure of the measure as one zooms in towards a point. Unlike classical methods based on the Fourier transform, our approach allows to establish normality in some non-integer bases and is robust under smooth perturbations of the measure. As applications, we complete and extend results of B. Host and E. Lindenstrauss on normality of ×p invariant measures, and many other classical normality results. This is a joint work with M. Hochman. 2.45 - 3.15: Coffee break 3.15 - 4.15: Károly Simon (Budapest) Title: Entropy of hidden Markov chain and the singularity of the Blackwell measure. Abstract: In 1957 the entropy of Hidden Markov chains was expressed by a measure which is called now the Blackwell measure. We give an upper bound on the Hausdorff dimension of the Blackwell measure, compute its entropy numerically for some models and compute numerically the entropy of the hidden Markov chain for an important example.

After the last talk we will go for drinks and possibly also dinner.