RENORMALIZATION TOWERS AND THEIR FORCING

摘要

A cyclic permutation pi : {1, ... , N} -> {1, ... , N} has a block structure if there is a partition of {1, ... , N} into k is not an element of {1, N} segments of consecutive integers (blocks) of the same length, permuted by p; call k the period of this block structure. Let p(1) < center dot center dot center dot < p(s) be periods of all possible block structures on pi. Call the finite string (p(1)/1, p(2)/p1, ... , 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 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 8 10 5 center dot center dot center dot 4n 4n+2 2n+1 center dot center dot center dot 2 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.