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A real number α is said to be b-normal if every m-long string of digits appears in the base-b expansion of α with limiting frequency b −m. We prove that α is b-normal if and only if it possesses no base-b " hot spot. " In other words, α is b-normal if and only if there is no real number y such that smaller and smaller neighborhoods of y are visited by the(More)
Rotation Theory has its roots in the theory of rotation numbers for circle homeomorphisms, developed by Poincaré. It is particularly useful for the study and classification of periodic orbits of dynamical systems. It deals with ergodic averages and their limits, not only for almost all points, like in Ergodic Theory, but for all points. We present the(More)
Combinatorial Dynamics has its roots in Sharkovsky's Theorem. This beautiful theorem describes the possible sets of periods of all cycles of a continuous map of an interval (or the real line) into itself. Here by a cycle I mean a periodic orbit, and by a period its minimal period. Consider the following Sharkovsky's ordering < s of the set N of natural(More)
We consider actions of the free semigroup with two generators on the real line, where the generators act as affine maps, one contracting and one expanding , with distinct fixed points. Then every orbit is dense in a half-line, which leads to the question whether it is, in some sense, uniformly distributed. We present answers to this question for various(More)