English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The goal of this paper is to analyze two measure preserving transformation ofcombinatorial and number-theoretical origin from the point of view of ergodicorbit theory. We study the Morse transformation (in its adic realization in thegroup $\textbf{Z}_2$ of integer dyadic numbers, as described by the author[{\sl J. Sov. Math.} {\textbf{28}}, 667-674 (1985); {\sl St. Petersburg Math.J.} {\textbf{6}} (1995), no. 3, 529-540]) and prove that it has the same orbitpartition as the dyadic odometer. Then we give a precise description of timesubstitution of the odometer, which produces the Morse transformation. It isconvenient to describe this time substitution in the form of randomre-orderings of the group $\mathbb Z$, or in terms of random infinitepermutations of the group $\mathbb Z$. We introduce the notion of {\it locallyfinite permutations (LFP) or locally finite bijection (LFB), and uniformlylocally finite time substitution (ULFTS)} for the group $\mathbb Z$ (and forall amenable groups). Two automorphisms which have the same orbit partitionsare called {\it allied} if the time substitution of one to another is ULFTS.Our main result is that the Morse transformation and the odometer are allied.The theory of random infinite permutations on the group $\mathbb Z$ (and onmore general groups) is as strong as the ergodic theory of actions of thegroup. The main task in this area is the investigation of infinitepermutations, and measures on the space of infinite permutations, as well asthe study of linear orderings on $\mathbb Z$. The class of locally finitepermutations is a useful class for such an analysis.

Orbit theory, locally finite permutations and morse arithmetic