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.