Renormalization towers and their forcing

Abstract:

A cyclic permutation $\pi :\{1, \dots , N\}\to \{1, \dots , N\}$ has a block structure if there is a partition of $\{1, \dots , N\}$ into $k\notin \{1,N\}$ segments of consecutive integers (blocks) of the same length, permuted by $\pi$; call $k$ the 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 renormalization tower of $\pi$. The same terminology can be used for patterns, i.e., for families of cycles of interval maps inducing the same, up to the flip of the entire orbit, cyclic permutation (thus, there are two permutations, one of whom is a flip of the other one, that define a pattern). A renormalization tower $\mathcal M$ 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: $4\gg 6\gg 3\gg 8\gg 10\gg 5\gg \dots \gg 4n\gg 4n+2\gg 2n+1\gg \dots \gg 2\gg 1$ understood in the strict sense (we write consecutive even numbers, starting with 4, then insert $m$ after each number of the form $2(2s+1)$, and finally append the order with 2 and 1). 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 $T$.