Existence of Infinite Orbits

This chapter begins Part 5 of the book. This part is devoted mostly to the study of the distribution of the plaid polygons: their size and number depending on the parameter. Section 21.2 gives a criterion for a point in FX to have a well-defined orbit. Section 21.3 revisits the pixelated spacetime diagrams of capacity 2, and uses them to construct a large supply of plaid polygons having large diameter. The construction in Section 21.3 works one parameter at a time. Section 21.4 takes the limit of our construction relative to a sequence of even rational parameters converging to our irrational parameter. This limiting argument completes the proof. Section 21.5 explains how to associate a plaid path to an infinite orbit. Section 21.6 gives a quick alternate proof of Theorem 21.1, based on results from [S1].

The Lerch zeta-function with algebraic irrational parameter

In this note, we present probabilisticlimit theorems on the complex plane as well as in functional spaces for the Lerch zeta-function with algebraic irrational parameter.

PROBABILISTIC ASPECTS OF EXTREME EVENTS GENERATED BY PERIODIC AND QUASIPERIODIC DETERMINISTIC DYNAMICS

We consider the distribution of the maximum for finite, deterministic, periodic and quasiperiodic sequences, and contrast the extreme value distributions in these cases with the classical results for iidrv's. A significant feature in the case of deterministic sequences is a multi-step structure for the distribution function. The extreme value distribution for the circle map with an irrational parameter is obtained in closed form with the help of the three-gap theorem for the map Xj+1 = (Xj + a) mod 1 where a ∈ (0,1) is an irrational number.