Abstracts

Valérie Berthé

Title: Multidimensional continued fractions and symbolic codings of toral translations

Abstract: It has been a long standing problem to find good symbolic codings for Kronecker toral translations that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. We construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any strongly convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings and bounded remainder sets at all scales in a natural way. Such sets provide particularly strong convergence properties of ergodic sums, and are also closely related to the notion of balance in word combinatorics. As strong convergence of a continued fraction algorithm results in a Pisot type property, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of substitutions. This is joint work with W. Steiner and J. Thuswaldner.

Keywords:return times, Poincaré recurrence theorem, minimal dynamical system, substitution subshift, odometer, continued fraction expansion.

Slides (PDF)
 

Jon Chaika

Title: There exists a weakly mixing billiard in a polygon

Abstract: This main result of this talk is that there exists a billiard flow in a polygon that is weakly mixing with respect to “Lebesgue” measure (on the unit tangent bundle to the billiard). This strengthens Kerckhoff, Masur and Smillie’s result that there exists ergodic billiard flows in polygons. This is joint work with Giovanni Forni.

Key words: weak mixing, Koopman operator, Baire Category Theorem.

Slides (PDF)
 

Vincent Delecroix

Title: An entropy upper bound for infinite interval exchange transformations

Abstract: Interval exchange transformations are maps obtained by cutting the interval and reassembling the pieces by translation in a different order. In the usual setting the unit interval is cut in finitely many pieces. In the infinite case, we cut in infinitely many pieces but require that the domain is a union of open subintervals of full measure. An infinite interval exchange transformation is a measurable automorphism with respect to the Lebesgue measure. It is rather immediate that finite interval exchange transformations have zero entropy. On the other hand, as proved by Arnoux-Ornstein-Weiss, any measurable automorphism can be realized as an infinite interval exchange transformation. In particular the entropy can be positive, even infinite. We present a refinement of a result of F. Blume that provides an upper bound on the entropy of such a map that depends only on the “cut part” of the transformation; that is the set of sizes of the subintervals in the domain.

Keywords: infinite interval exchange transformations, entropy, cutting and stacking.

Slides (PDF)
 

Fabien Durand

Title: Self induced systems

Abstract: In this talk we will characterize minimal Cantor systems and ergodic systems \((X,T)\) for which there exists some subset \(U\) of \(X\) having a well defined return map \(T_U:U–>U\) such that \((X,T)\) is isomorphic to \((U, T_U)\).

Keywords: return times, Poincaré recurrence theorem, minimal dynamical systems, substitution subshifts, odometers, Bratteli diagrams.

Slides (PDF)
 

Jeremias Epperlein

Title: Iterated Minkowski sums, horoballs and north-south-dynamics

Abstract: Let \(G\) be a countable group generated by a finite set \(A\) containing the identity. This talk will present results about the topological dynamical system \(\varphi_A: 2^G \to 2^G\) defined by \(M \mapsto MA=\{ma : m \in M, a \in A\}\). I will concentrate on properties of the pair \((G,A)\) which can be recovered from the dynamics of \(\varphi_A\). This is based on joint work with Tom Meyerovitch.

Keywords: amenable group, topological conjugacy, topological factor map.

Slides (PDF)
 

Florian Richter

Title: An analogue of Furstenberg’s sumset conjecture in the integers

Abstract: Using the language of fractal geometry and dynamical systems, in the late 1960’s Hillel Furstenberg proposed a series of conjectures that explore the relationship between digit expansions in distinct prime bases. While his famous x2-x3 conjecture in ergodic theory remains unsolved, recent solutions to his equally renowned “transversality” conjectures have shed new light on old problems. While there is a strong historical precedent for this inquiry on the real line, less seems to be known in the discrete setting of the integers. In this talk, we investigate transversality in the integers with the aim of understanding the independence of sets of integers that are multiplicatively structured with respect to different bases. This is based on joint work with Daniel Glasscock and Joel Moreira.

Slides (PDF)