Rigidity and Non-recurrence along Sequences

TL;DR

This paper investigates the structural properties of sequences that induce rigidity or non-recurrence in dynamical systems, aiming to classify sequences based on their behavior in weakly mixing and ergodic transformations.

Contribution

It explicitly characterizes the sequences that can serve as rigidity or non-recurrent sequences for certain classes of dynamical systems, revealing their complex structure.

Findings

Identifies classes of sequences that can be rigidity sequences.

Identifies classes of sequences that can be non-recurrent sequences.

Shows the rich and complicated structure of these sequences.

Abstract

Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences.