# Numbers in Ergodic Theory

This year there will be a series of one day meetings taking place in Leiden and Utrecht on the topic of
'Numbers in Ergodic Theory'. 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 Charlene Kalle (kallecccj@math.leidenuniv.nl) and Tom Kempton (T.M.W.Kempton@uu.nl), please feel free to email one of us if you have any questions.

The meetings are supported by an 'Incidentele Steun' grant of the NWO and a seminar grant from STAR.

## 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. |
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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.' |
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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 (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 functionβ ↦ 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). |
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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. |
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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. |
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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. |
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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. |
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12.30 - 1.45: | Lunch | |

1.45 - 2.45: | Karma Dajani (Utrecht) |
Title: Optimal Expansions in non-integer base |

Abstract: click here. |
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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 |
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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. |
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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.