Renormalization towers and their forcing

A cyclic permutation $pi:{1, dots, N}to {1, dots, N}$ has a emph{block structure} if there is a partition of ${1, dots, N}$ into $k otin{1,N}$ segments (emph{blocks}) permuted by $pi$; call $k$ the emph{period} of this block structure. Let $p_1<dots <p_s$ be periods of all possible block structures on $pi$. Call the finite string $(p_1/1,$ $p_2/p_1,$ $dots,$ $p_s/p_{s-1}, N/p_s)$ the {it renormalization tower of $pi$}. The same terminology can be used for emph{patterns}, i.e., for families of cycles of interval maps inducing the same (up to a flip) cyclic permutation. A renormalization tower $mathcal M$ emph{forces} a renormalization tower $mathcal N$ if every continuous interval map with a cycle of pattern with renormalization tower $mathcal M$ must have a cycle of pattern with renormalization tower $mathcal N$. We completely characterize the forcing relation among renormalization towers. Take the following order among natural numbers: $ 4gg 6gg 3gg dots gg 4ngg 4n+2gg 2n+1ggdots gg 2gg 1 $ understood in the strict sense. We show that the forcing relation among renormalization towers is given by the lexicographic extension of this order. Moreover, for any tail $T$ of this order there exists an interval map for which the set of renormalization towers of its cycles equals $T$.