Chaos, cycles and orbit distributions for families of variants of the shift map

Abstract

We define dynamical systems of families of maps on the unit interval. The mapś, called variants, are obtained by complementation of selected linear segments of the Bernoulli Shift. We determine a recursive relation for cycle structure. We show that if x₀ is normal to base m then the orbit of x₀ under a symmetric variant of B_m is uniformly distributed. We utilize the fact that inverse images of open sets under a symmetric variant are symmetric sets, and on symmetric sets the diicrete time average for symmetric variants and the shift are equal. For non-symmetric variants we exhibit an inequality for the discrete time average of iterates of normal numbers which is based on the number of intervals over which f is non-symmetric

Keywords:

• discrete time dynamical system
• periodic point
• periodic orbit
• orbit distribution