# Ergodic Theory and Fractal Days

This semester there will be three one day meetings taking place in Utrecht, Leiden and Delft on the topic of
ergodic theory and fractals. 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 organised by Karma Dajani (k.dajani1@uu.nl), Charlene Kalle (kallecccj@math.leidenuniv.nl) and Cor Kraaikamp (c.kraaikamp@tudelft.nl), please feel free to email one of us if you have any questions.

The meetings are supported by a seminar grant from STAR.

## 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.