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

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 48 in order to determine those with maximal entropy, therefore making it a feasible problem.