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On conjugacy of natural extensions of one-dimensional maps

Part of: Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  20 September 2022

J. BOROŃSKI Open the ORCID record for J. BOROŃSKI [Opens in a new window] ,
P. MINC  and
S. ŠTIMAC
J. BOROŃSKI
Affiliation:
National Supercomputing Centre IT4Innovations, University of Ostrava, IRAFM, 30. dubna 22, 70103 Ostrava, Czech Republic (e-mail: jan.boronski@osu.cz)
P. MINC
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA (e-mail: mincpio@auburn.edu)
S. ŠTIMAC*
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia
*
e-mail: sonja@math.hr
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Abstract

We prove that for any non-degenerate dendrite D, there exist topologically mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$ such that the natural extensions (as known as shift homeomorphisms) $\sigma _F$ and $\sigma _f$ are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map f does not depend on the dendrite D and can be selected so that the inverse limit $\underleftarrow {\lim } (D,F)$ is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and Hénon maps to natural extensions of dendrite maps.

Keywords

natural extensioninverse limitdendriteintervalpseudo-arc

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37B45: Continua theory in dynamics
Secondary: 37D45: Strange attractors, chaotic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 9 , September 2023 , pp. 2915 - 2937
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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