Irregular model sets and tame dynamics

Abstract

We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces.In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides negative answers to questions by Schlottmann and Moody in the Euclidean setting.

Minimal systems with finitely many ergodic measures

“…Since π ′ is N to one, we conclude that the number of the ergodic measures of X ′ is bounded by N(k − 1) N , and so does X , ending the proof. We mention that to obtain Proposition 3.7 we need to use some result (Proposition 3.2) which is a generalization of the previous one obtained by Fuhrmann, Glasner, Jäger and Oertel [11].…”

“…Recently, a striking result proved by Fuhrmann, Glasner, Jäger and Oertel solved a long open question, i.e. the authors showed that a minimal tame system is regular [11].…”

“…(1) In [11,Subsection 5.3], it was shown that there is a minimal system which is an at most two-to-one and almost one to one extension of its maximal equicontinuous factor, but exhibits two distinct ergodic invariant measures. This system has 2-ITtuples but no 3-IT-tuples.…”

“…If m(Y N ) = 1, then we say that π is a regular almost N to 1 map, where m is the Haar measure on X eq . In [11], it was shown that a minimal tame system is regular. Question 1.…”

On entropy of Delone sets of p-adic numbers

“…In [BLR07, Theorem 1] it is shown that for FLC Delone sets ω in R d the patch counting entropy E pc (ω) equals the topological entropy E(π ω ) of this associated dynamical system. This observation is used for example in [PFS14,FGJO18] in order to define the entropy of FLC Delone sets. In [BLR07] the equivalence of these approaches is used to show that FLC Delone sets with "pure point dynamical spectrum" have 0 patch counting entropy.…”