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LI–YORKE CHAOS ALMOST EVERYWHERE: ON THE PERVASIVENESS OF DISJOINT EXTREMALLY SCRAMBLED SETS

Part of: Low-dimensional dynamical systems Mathematical economics Applications - Dynamical systems and ergodic theory

Published online by Cambridge University Press:  03 March 2022

LIUCHUN DENG Open the ORCID record for LIUCHUN DENG [Opens in a new window] ,
M. ALI KHAN Open the ORCID record for M. ALI KHAN [Opens in a new window]  and
ASHVIN V. RAJAN Open the ORCID record for ASHVIN V. RAJAN [Opens in a new window]
LIUCHUN DENG*
Affiliation:
Social Sciences Division, Yale-NUS College, Singapore
M. ALI KHAN
Affiliation:
Department of Economics, The Johns Hopkins University, Baltimore, MD 21218, USA e-mail: akhan@jhu.edu
ASHVIN V. RAJAN
Affiliation:
3935 Cloverhill Road, Baltimore, MD 21218, USA e-mail: ashvinrj@aol.com
*
e-mail: yncdl@nus.edu.sg
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Abstract

We show that there exists a continuous function from the unit Lebesgue interval to itself such that for any $\epsilon \geq 0$ and any natural number k, any point in its domain has an $\epsilon $ -neighbourhood which, when feasible, contains k mutually disjoint extremally scrambled sets of identical Lebesgue measure, homeomorphic to each other. This result enables a satisfying generalisation of Li–Yorke (topological) chaos and suggests an open (difficult) problem as to whether the result is valid for piecewise linear functions.

Keywords

scrambled setextremally scrambled setSmith–Volterra–Cantor setLebesgue measureLi–Yorke chaostopological chaos

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37N40: Dynamical systems in optimization and economics 91B55: Economic dynamics 92D25: Population dynamics (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 132 - 143
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Liuchun Deng acknowledges the support of a start-up grant from Yale-NUS College.

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