Advances and Applications in Discrete Mathematics
Volume 14, Issue 2, Pages 95 - 123
(October 2014)
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VARIATION OF POINCARÉ RECURRENCE TIMES IN DISCRETE DYNAMICAL SYSTEMS
Mimoon Ismael, Rodney Nillsen and Graham Williams
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Abstract: This paper is concerned with the variation and standard deviation of recurrence times in discrete dynamical systems. The first sections of the paper deal with dynamical systems where f acts as a cyclic permutation on a finite set S. When A is a subset of S, an expression is derived for the standard deviation of the recurrence times of points in A, and the maximum and minimum of are derived over sets A in the family of subsets of S that have r gaps and s elements. The results are applied to the case of more general discrete systems obtained as a countable “sum” of finite systems. Various aspects are discussed, and interpreted geometrically in some cases. Whereas the average value of the recurrence time is finite, the standard deviation of the recurrence time in some of these systems may be infinite. |
Keywords and phrases: discrete systems, recurrence times, Kac’s formula, standard deviation of recurrence times, sums of discrete systems. |
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