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

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 91]. 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 . The obtained results agree with the conjectured ones. The conjecture that we can restrict our attention to C ^* _n is motivated theoretically. On the other hand, it is worth noticing that the complexity of examining all cycles in C ^* ₂₂, C ^* ₂₆, and C ^* 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.

Keywords:

• 37B40, 37E15, 37M99
• Combinatorial dynamics
• interval map
• topological entropy
• cycles of maximum entropy