Decomposition Towers and their Forcing


March 23, 2018 | 2:30 - 3:30 p.m.


Campbell Hall 443


Alexander Blokh, UAB and Michal Misiurewicz, IUPUI, Indianapolis


We define the decomposition tower, a new characteristic of cyclic permutations. A cyclic permutation π of the set N = {1,…,n} has a block structure if N can be divided into consecutive blocks permuted by π. The set N might be partitioned into blocks in a few ways; then those partitions get finer and finer. Decomposition towers reflect the variety of sizes of blocks of such partitions. Set

4 >> 6 >> 3 >> … >> 4n >> 4n + 2 >> … >> 2 >> 1,

define the lexicographic extension of >> onto towers, and denote it >> too. We prove that if N >> M and an interval map f has a cycle with decomposition tower N then f must have a cycle with decomposition tower M. The results are joint with Michal Misiurewicz (IUPUI, Indianapolis), inspired by the Sharkovsky Theorem, and based upon our (M – B) recent results.