Renormalization towers and their forcing
Alexander Blokh and Michał Misiurewicz
Abstract
A cyclic permutation
π:{1, 2, …, n}→{1, 2, …, n} has a
block structure if there is a partition of
{1, 2, …, n} into k∉{1,n} segments
(blocks) permuted by π; call k the period of
this block structure.
Let p1 < … < ps be periods of all
possible block structures on π. Call the finite string
(p1/1, p2/p1, …, ps/ps-1, n/ps)
the renormalization tower of π. The same terminology can be
used for patterns, i.e., for families of cycles of interval
maps inducing the same (up to a flip) cyclic permutation. A
renormalization tower M forces a renormalization tower N
if every continuous interval map with a cycle of pattern with
renormalization tower M must have a cycle of pattern with
renormalization tower N. We completely characterize the forcing
relation among renormalization towers. Take the following order among
natural numbers: 4 ≫ 6 ≫ 3 ≫
… ≫ 4n ≫ 4n+2 ≫
2n+1 ≫ 4n+4 ≫ … ≫ 2 ≫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.