Structure and dimension of invariant subsets of expanding Markov maps and joint invariance
(2023) In Dynamical Systems 38(3). p.405-426- Abstract
A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyse the structure of the invariant subsets of a single map. For a mapping of the circle of class (Formula presented.), (Formula presented.), we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one... (More)
A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyse the structure of the invariant subsets of a single map. For a mapping of the circle of class (Formula presented.), (Formula presented.), we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one non-trivial joint invariant compact subset. Lastly we show the strong dimensional relation between the maximal invariant subset of a given Markov map contained in a subinterval (Formula presented.) and the set of all right endpoints of its invariant subsets that are contained in the same subinterval, (Formula presented.), as well as the continuous dependence of the dimension on the endpoints of the subinterval (Formula presented.).
(Less)
- author
- Lamprinakis, Georgios LU
- organization
- publishing date
- 2023
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Dimension theory, expanding Markov maps, joint invariant subset, symbolic dynamics, β-expansion
- in
- Dynamical Systems
- volume
- 38
- issue
- 3
- pages
- 405 - 426
- publisher
- Taylor & Francis
- external identifiers
-
- scopus:85151938148
- ISSN
- 1468-9367
- DOI
- 10.1080/14689367.2023.2194520
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
- id
- 4a03371c-0b56-4c47-bf19-089caa1da9ba
- date added to LUP
- 2023-07-20 11:49:55
- date last changed
- 2023-10-26 14:46:45
@article{4a03371c-0b56-4c47-bf19-089caa1da9ba,
abstract = {{<p>A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyse the structure of the invariant subsets of a single map. For a mapping of the circle of class (Formula presented.), (Formula presented.), we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one non-trivial joint invariant compact subset. Lastly we show the strong dimensional relation between the maximal invariant subset of a given Markov map contained in a subinterval (Formula presented.) and the set of all right endpoints of its invariant subsets that are contained in the same subinterval, (Formula presented.), as well as the continuous dependence of the dimension on the endpoints of the subinterval (Formula presented.).</p>}},
author = {{Lamprinakis, Georgios}},
issn = {{1468-9367}},
keywords = {{Dimension theory; expanding Markov maps; joint invariant subset; symbolic dynamics; β-expansion}},
language = {{eng}},
number = {{3}},
pages = {{405--426}},
publisher = {{Taylor & Francis}},
series = {{Dynamical Systems}},
title = {{Structure and dimension of invariant subsets of expanding Markov maps and joint invariance}},
url = {{http://dx.doi.org/10.1080/14689367.2023.2194520}},
doi = {{10.1080/14689367.2023.2194520}},
volume = {{38}},
year = {{2023}},
}