On the partitions with Sturmian-like refinements

TL;DR

This paper investigates the evolution of certain partitions in circle rotation dynamics, revealing their eventual connection to Sturmian partitions and implications for Sturmian subshifts and factor mappings.

Contribution

It demonstrates that partitions with disconnected atoms under irrational rotation eventually refine into connected intervals related to Sturmian partitions, linking dynamics and symbolic coding.

Findings

Refinements of partitions eventually match Sturmian preimages

Partitions' atoms become connected intervals over time

Injectivity of large block codes in Sturmian subshifts

Abstract

In the dynamics of a rotation of the unit circle by an irrational angle $\alpha\in(0,1)$, we study the evolution of partitions whose atoms are finite unions of left-closed right-open intervals with endpoints lying on the past trajectory of the point $0$. Unlike the standard framework, we focus on partitions whose atoms are disconnected sets. We show that the refinements of these partitions eventually coincide with the refinements of a preimage of the Sturmian partition, which consists of two intervals $[0,1-\alpha)$ and $[1-\alpha,1)$. In particular, the refinements of the partitions eventually consist of connected sets, i.e., intervals. We reformulate this result in terms of Sturmian subshifts: we show that for every non-trivial factor mapping from a one-sided Sturmian subshift, satisfying a mild technical assumption, the sliding block code of sufficiently large length induced by the mapping is injective.