On the Invertibility and Structural Properties of Interval Translation Maps

In this paper, we present a significant advancement by proving that each Interval Translation Map (ITM) is invertible almost everywhere on the support of its invariant measure. This result has far-reaching consequences, as it implies that ITMs reduce to Interval Exchange Maps (IEM) in some cases. Wh...

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Published inLobachevskii journal of mathematics Vol. 46; no. 2; pp. 901 - 910
Main Authors Tajbakhsh, Khosro, Yaghmaeian, Reza
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.02.2025
Springer Nature B.V
Subjects
Algebra
Analysis
Geometry
Invariants
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
invertibility
interval translation maps
syndetic recurrence
interval exchange maps
37E10
invariant measure
37A05
37E05
Online AccessGet full text
ISSN1995-0802
1818-9962
DOI10.1134/S1995080224606891

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Summary:In this paper, we present a significant advancement by proving that each Interval Translation Map (ITM) is invertible almost everywhere on the support of its invariant measure. This result has far-reaching consequences, as it implies that ITMs reduce to Interval Exchange Maps (IEM) in some cases. While previous work has noted that the support of the invariant measure may sometimes form a Cantor set, we are able to conclude that every ITM is indeed invertible almost everywhere on its support. This discovery extends the understanding of ITMs and their structural properties, providing a deeper insight into the behavior of these dynamical systems. Additionally, we investigate several properties of ITMs, contributing to a deeper understanding of their structural properties.
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ISSN:1995-0802
1818-9962
DOI:10.1134/S1995080224606891