Rotation numbers for invariant sets under zⁿ on the unit circle
by
James M. Malaugh
University of Alabama at Birmingham
Coauthors: Alexander M. Blokh, John C. Mayer, Lex G. Oversteegen, Daniel K. Parris

Consider the unit circle T = R/Z. The map \sigma_n:T --> T is defined by \sigma_n(x) = nx mod 1. For n=2, the minimal invariant sets A subset T with rotation number under the map have been described topologically and combinatorially by Bullet and Sentenac. For example, it is known that any invariant set with rotation number must lie within a closed semicircle, and that the graph from semicircle parameter to rotation number is a (1-dimensional) Devil's staircase. What differences arise when we consider n > 2? A theorem that lays the groundwork for studying these differences is the following:

Theorem \sigma_n, for n >= 2, is order-preserving on A subset T iff T - A contains n-1 disjoint, open intervals (called ``gaps''), each of length 1/n.

Using the location of the gaps as parameters, we define an (n-1)-dimensional parameter space \Delta_n of maps, each of which is the continuous monotone extension of the restriction of \sigma_n to the complement of the gaps, corresponding to those invariant sets with rotation numbers under \sigma_n. We define a rotation number function \rho_n:\Delta_n --> [0, 1]. For this talk, we describe the properties \rho₃. In particular, we have

Theorem The graph of \rho₃ is a 2-dimensional Devil's staircase over \Delta₃.

We also describe the corresponding minimal invariant sets which have rotation numbers under \sigma₃.

Date received: February 9, 2001