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2008 A Lower Bound for the Maximum Topological Entropy of $(4k+2)$-Cycles
Lluís Alsedà, David Juher, Deborah M. King
Experiment. Math. 17(4): 391-408 (2008).

Abstract

For continuous interval maps we formulate a conjecture on the shape of the cycles of maximum topological entropy of period $4k+2$ We also present numerical support for the conjecture. This numerical support is of two different kinds. For periods $6$, $10$, $14$, and $18$ we are able to compute the maximum-entropy cycles using nontrivial ad hoc numerical procedures and the known results of Jungreis, 1991. In fact, the conjecture we formulate is based on these results.

For periods $n=22$, $26$, and $30$ we compute the maximum-entropy cycle of a restricted subfamily of cycles denoted by $C_n^\ast$. The obtained results agree with the conjectured ones. The conjecture that we can restrict our attention to $C_n^\ast$ is motivated theoretically. On the other hand, it is worth noticing that the complexity of examining all cycles in $C^\ast_{22}$, $C^\ast_{26}$, and $C^\ast_{30}$ is much less than the complexity of computing the entropy of each cycle of period $18$ in order to determine those with maximal entropy, therefore making it a feasible problem.

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Lluís Alsedà. David Juher. Deborah M. King. "A Lower Bound for the Maximum Topological Entropy of $(4k+2)$-Cycles." Experiment. Math. 17 (4) 391 - 408, 2008.

Information

Published: 2008
First available in Project Euclid: 27 May 2009

zbMATH: 1182.37015
MathSciNet: MR2484424

Subjects:
Primary: 37B40 , 37E15 , 37M99

Keywords: Combinatorial dynamics , cycles of maximum entropy , interval map , topological entropy

Rights: Copyright © 2008 A K Peters, Ltd.

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