Dynamics beyond the Pisot Condition

In Short

What? Thematic Programme

When? May 10, 2027 — June 25, 2027

Where? University of Wien, Erwin Schrödinger Institute, ESI Boltzmann Lecture Hall

Description

The theory of zero entropy dynamical systems covers a vast ground, in which symbolic dynamics, polygonal billiards & piecewise isometries, and numeration systems are but three yet active areas. This thematic semester is inspired by three interconnected areas where we think future efforts of these communities will veer to.

1) There is a vast literature on Pisot substitutions and their dynamics and although the dynamics of Pisot substitutions is well-understood, the famous Pisot conjecture on pure discreteness of the spectrum of Pisot substitutive shifts is still open. Particularly interesting objects in the context of Pisot dynamics are the so-called Rauzy fractals, that reflect the dynamics of Pisot substitutive shifts geometrically.

Our aim is to go beyond the case of Pisot substitutions, where the picture is much more incomplete. One way to generalize the Pisot substitution setting is to consider sequences of substitutions instead of a single substitution.

2) Multi-dimensional continued fraction algorithms (sometimes called subtractive algorithms, find their roots in trying to find approximations of real vectors by rational vectors with a common denominator. Similar algorithms appear in a variety of applications, such as Rauzy-Veech induction of IETs.

Well-known multi-dimensional continued fraction algorithms, such as Jacobi-Perron, Brun and Selmer get increasingly complicated as the dimension grows; beyond a certain dimension threshold their second eigenvalue is no longer inside the unit disk, so that difficulties similar to non-Pisot substitutions arise.

3) Interval Exchange Transformations (IETs) emerged in the 1970 by Katok, Keane and others in the 1970, and by now is very far developed. Infinite Interval Exchange Transformations consists of permutations of infinitely many (as opposed to finitely many for IETs) intervals. They can serve as a genera model for Cantor systems and most naturally as Poincare maps of flow on infinite genus surfaces. However, many of the basic questions about minimality, (unique) ergodicity and (weak) mixing are still largely open.

Organizers

Ayreena Bakhtawar (IMPAN, Warsaw)

Henk Bruin (University of Vienna)

Jörg Thuswaldner (Montanuniversität Leoben)

Funding