Ergodic properties of countable extensions

<p>First, we study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to an interval map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give examples, both Markov and non-Markov, for which the criterion is violated. ^ Next, we establish a criterion for the existence of a constant slope map on the extended real line conjugate to a given countably piecewise monotone interval map. We require the given interval map to be continuous, Markov, and topologically mixing, and show by example that the mixing hypothesis is essential. ^ Next, we study a class of countable state subshifts of finite type which admit finite-state factors. Our systems carry a displacement function, analogous to that used in the rotation theory of circle maps. Among those invariant measures on the factor system for which the average displacement is zero, we identify a unique measure of maximal entropy. As a corollary we obtain an efficient computational tool for the Gurevich entropy of the countable state system. We also prove that the countable state systems in our class do not admit any measure of maximal entropy. ^ Finally, we apply our findings to the study of degree one circle maps with Markov partitions and with transitive liftings to the real line. After compactifying by adjoining fixed points at plus and minus infinity, we show how to compute the topological entropy of the lifting and how to find all conjugate maps of constant slope. We prove that there are conjugate maps of constant slope for every slope greater than or equal to the exponential of the entropy.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2015

Author dc:creator

• Roth, Samuel Joshua

Contributors dc:contributor

• Michal Misiurewicz
• Alexandre Eremenko
• Roland Roeder
• Rodrigo Perez

• Mathematics

Repository record dc:identifier

https://docs.lib.purdue.edu/open_access_dissertations/553

OAI identifier oai:identifier

oai:docs.lib.purdue.edu:open_access_dissertations-1500