Topological prevalence of finite type maps and density of stability for interval translation mappings
File(s)
Author(s)
Staresinic, Leon
Type
Thesis
Abstract
An Interval Translation Mapping (ITM) is a piece-wise translation T : I → I defined on a finite partition I1, . . . , Ir of an interval I into r g 2 subintervals. We do not assume that the images of these intervals are disjoint. These maps naturally generalize the classical Interval Exchange Transformations (IETs) by removing the bijectivity assumption. Let ITM(r) be the space of all Interval Translation Mappings, where we fix r but not the intervals I1, . . . , Ir, nor the translations on each of them. The set X(T) := T ng0 T n (I) is then either a finite union of intervals (and T behaves like an IET on those intervals), in which case the map is called of finite type, or is a disjoint union of finitely many intervals and a Cantor set, in which case the map is called of infinite type. In Chapter III, for an arbitrary r g 2, we show that the set of finite type maps contains an open and dense set of ITM(r). This resolves in positive a topological version of a long standing conjecture due to Boshernitzan and Kornfeld. More precisely, we show that there exists an open and dense subset S(r) of ITM(r) consisting of stable maps such that each T ∈ S(r): - is of finite type; - the first return map to any component of X(T) corresponds to a circle rotation; - S(r) ∋ T 7→ X(T) is continuous in the Hausdorff topology. In Chapters I and II we give an introduction to and an overview of the fields of IETs and ITMs, respectively.
Version
Open Access
Date Issued
2025-06-30
Date Awarded
01/10/2025
License URL
Advisor
van Strien, Sebastian
Drach, Kostiantyn
Sponsor
Department of Mathematics
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)