Soo, TerryNualart, DavidWilkens, Amanda2024-06-292024-06-292021-05-312021http://dissertations.umi.com/ku:17730https://hdl.handle.net/1808/35240In ergodic theory, where measure-preserving dynamical systems are common objectsof study, the entropy of a system can sometimes be used to determine if two systems are isomorphic. In this dissertation, we construct factor maps for two systems, influenced by entropic results, with methods from the fields of probability and symbolic dynamics. First, we construct entropy increasing monotone factors in the context of a Bernoulli shift over the free group of rank at least two. Second, we give an elementary construction of an isomorphism between Poisson point processes over Rd, acted on by the group of isometries of Rd, that is finitary. In both cases, we build on the groundbreaking work of Ornstein and Weiss in their 1987 paper [22]. As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss proved that any two Poisson point processes are isomorphic as factors. Via our elementary construction of such an isomorphism, we are able to add the finitary condition. Ornstein and Weiss further provided the first example of an entropy increasing factor map for Bernoulli shifts over the free group of rank two. We modify this example to gain the monotone condition. The material included in Chapters 2 and 3 of this dissertation may be found in [30] and [29], respectively. Both works are joint between Terry Soo and the author.76 pagesenCopyright held by the author.MathematicsBernoulli shiftdynamical systemsergodic theoryPoisson point processprobabilityDissertation