Renormalization towers and their forcing

Abstract

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 $k\notin \{1,N\}$ segments (\emph{blocks}) permuted by $\pi$; call $k$ the \emph{period} of this block structure. Let $p_1<\cdots <p_s$ 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: $4\gg 6\gg 3\gg \cdots \gg 4n\gg 4n+2\gg 2n+1\gg \cdots \gg 2\gg 1$ 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 $T$.

arXiv comment19 pages, 5 figures