Supervisors info:
Δημήτριος Γατζούρας, Καθηγητής, Τμήμα Μαθηματικών, Σχολή Θετικών Επιστημών
Summary:
In the first chapter of this thesis, we study the concept of recurrence in topological dynamics and we give some applications to diophantine inequalities, including a well-known result of Hardy and Littlewood and a generalization of polynomials, some combinatorial applications and we prove Hilbert's theorem.
In the second chapter we study the concept of multiple recurrence for a finite set of commuting continuous maps into a compact metric space, we give the proof of van der Waerden's theorem through Birkhoff Multiple Recurrence theorem and the multidimensional van der Waerden's theorem (Grunwald theorem), we introduce the concept of IP-sets and finally we obtain some combinatorial applications and applications into diophantine inequalities, including another proof of the previous theorem of Hardy and Littlewood.
Finally, in the third chapter we study the concept of proximality in dynamic systems, central sets are defined and a proof of Hindman's theorem is given. There is also a proof of Schur's theorem, another proof of the van der Waerden's theorem, the introduction of IP-systems, and Schur and Brauer's generalization of the Schur and van der Waerden's theorems. Finally, a dynamic proof of the theorem of Hales and Jewett, of a (very strong) generalization of van der Waerden's theorem, is given.
Keywords:
Topological Dynamics, van der Waerden, Hales-Jewett,
