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Abstract:

Ecclestiastes 1.15 We are now in position to develop our rst prototype application of the periodic orbit theory: cycle counting. This is the simplest illustration of raison d'etre of the periodic orbit theory; we shall develop a duality transformation that relates local information- in this case the next admissible symbol in a symbol sequence- to global averages, in this case the mean rate of growth of the number of admissible itineraries with increasing itinerary length. We shall turn the topological dynamics of the preceding chapter into a multiplicative operation by means of transition matrices/Markov graphs, and show that the powers of a transition matrix count the distinct itineraries. The asymptotic growth rate of the number of admissible itineraries is therefore given by the leading eigenvalue of the transition matrix; the leading eigenvalue is given by the leading zero of the characteristic determinant of the transition matrix, which is in this context called the topological zeta function. For a class of ows with nite Markov graphs this determinant is a nite polynomial which can be read o the Markov graph. The method goes well beyond the problem at hand, and forms the core of the entire treatise, to be taken up again in chapter 10. 9.1 Counting itineraries In the 3-disk system the number of admissible trajectories doubles with every iterate: there are K n = 3 2 n distinct itineraries of length n. If there is pruning, this is only an upper bound and explicit formulas might be hard to come by, but we still might be able to establish a lower exponential bound of form K n Ce n ^ h

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