Barrueco, D. J. (2022). Ergodic Theory, Entropy, and x2,x3: An Exposition of Furstenberg’s x2,x3 Conjecture. Retrieved from https://doi.org/10.14418/wes01.2.344
In this thesis, we will discuss a branch of dynamics called ergodic theory, and topics within the branch such as symbolic spaces, invariant measures, and entropy theory. Ergodic theory is the study of dynamical systems with respect to measure – we tailor a discussion on its foundations (and those of measure theory) to lead into one on Furstenberg's x2,x3 problem. Furstenberg's conjecture states that if a Borel probability measure µ on [0,1) is invariant under the transformations T(x) := 2x (mod 1) and S(x) := 3x (mod 1), and ergodic under the action of the semigroup generated by those transformations, then it is either atomic or Lebesgue measure. The conjecture has not been proved, but Daniel Rudolph proved a weaker statement, imposing the additional assumption that µ have positive entropy with respect to one of T or S. We use our earlier discussions on ergodic theory and entropy theory to summarize Rudolph's work.