## Experimental Mathematics

- Experiment. Math.
- Volume 17, Issue 4 (2008), 391-408.

### A Lower Bound for the Maximum Topological Entropy of $(4k+2)$-Cycles

Lluís Alsedà, David Juher, and Deborah M. King

#### 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.

#### Article information

**Source**

Experiment. Math., Volume 17, Issue 4 (2008), 391-408.

**Dates**

First available in Project Euclid: 27 May 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1243429953

**Mathematical Reviews number (MathSciNet)**

MR2484424

**Zentralblatt MATH identifier**

1182.37015

**Subjects**

Primary: 37B40: Topological entropy 37E15: Combinatorial dynamics (types of periodic orbits) 37M99: None of the above, but in this section

**Keywords**

Combinatorial dynamics interval map topological entropy cycles of maximum entropy

#### Citation

Alsedà, Lluís; Juher, David; King, Deborah M. A Lower Bound for the Maximum Topological Entropy of $(4k+2)$-Cycles. Experiment. Math. 17 (2008), no. 4, 391--408. https://projecteuclid.org/euclid.em/1243429953