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Some results on joint ergodicity, sets of recurrence and substitution and tiling systems
Son, Younghwan

2013, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis consists of five parts. Chapter 1 is an introduction of the main problems and results that we will deal with in this thesis. Chapter 2 provides some preliminary notions and results in ergodic theory which will be used throughout this thesis.

In Chapter 3, we study jointly ergodic actions of abelian groups. We prove that jointly ergodic actions are totally jointly ergodic if each action is totally ergodic. This result reveals new properties of multi-parameter actions, improving upon the previous work of Berend on commuting jointly ergodic Z-actions.

In Chapter 4, we present some results on sets of recurrence and van der Corput sets in Z^k achieved in collaboration with Bergelson, Kolesnik, Madritsch and Tichy, which refine and unify some of the previous results obtained by Sarkozy, Furstenberg, Kamae and Mendes France, and Bergelson and Lesigne.

In Chapter 5, we investigate the substitution dynamical systems and the associated tiling dynamical systems arising from the substitutions θ: 0 → 001, 1 → 11001 and η: 0 → 001, 1 → 11100, which were studied by Kakutani, Dekking and Keane, and Berend and Radin. It was known before that substitution systems arising from θ and η are weakly mixing but not strongly mixing. Firstly, we improve this result by showing that these systems have minimal self-joinings and hence are mildly mixing. Secondly, we construct tiling dynamical systems which are mildly mixing.
Vitaly Bergelson (Advisor)
Alexander Leibman (Committee Member)
Nimish Shah (Committee Member)

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Son, Y. (2013). Some results on joint ergodicity, sets of recurrence and substitution and tiling systems. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Son, Younghwan. "Some results on joint ergodicity, sets of recurrence and substitution and tiling systems." Electronic Thesis or Dissertation. Ohio State University, 2013. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Son, Younghwan "Some results on joint ergodicity, sets of recurrence and substitution and tiling systems." Electronic Thesis or Dissertation. Ohio State University, 2013. https://etd.ohiolink.edu/

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