Asymptotic Formula for Counting in Deterministic and Random Dynamical Systems Page: I
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Naderiyan, Hamid. Asymptotic Formula for Counting in Deterministic and
Random Dynamical Systems. Doctor of Philosophy (Mathematics), May 2023, 82 pp., 40
numbered references.
The lattice point problem in dynamical systems investigates the distribution of
certain objects with some length property in the space that the dynamics is defined. This
problem in different contexts can be interpreted differently. In the context of symbolic
dynamical systems, we are trying to investigate the growth of N(T), the number of finite
words subject to a specific ergodic length T, as T tends to infinity. This problem has been
investigated by Pollicott and Urbanski to a great extent. We try to investigate it further,
by relaxing a condition in the context of deterministic dynamical systems. Moreover, we
investigate this problem in the context of random dynamical systems. The method for us
is considering the Fourier-Stieltjes transform of N(T) and expressing it via a Poincar6
series for which the spectral gap property of the transfer operator, enables us to apply
some appropriate Tauberian theorems to understand asymptotic growth of N(T). For
counting in the random dynamics, we use some results from probability theory.
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Naderiyan, Hamid. Asymptotic Formula for Counting in Deterministic and Random Dynamical Systems, dissertation, May 2023; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc2137552/m1/2/?rotate=270: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .