Τομέας Μαθηματικής Ανάλυσης
Library of the School of Science

2011-12-21

2011

Κουτσογιάννης Ανδρέας

Βασιλική Φαρμάκη Καθηγήτρια Επιβλέπουσα

Θέματα της εργοδικής θεωρίας Ramsey

Greek

In this Ph.D. thesis we are dealing with fundamental problems from Ramsey
theory and topological dynamics theory, highlighting the connection between
them.
In Chapter 2 we develop in a systematic way a Ramsey theory for words, in fact
ω-Ζ*-located words, over an infinite alphabet dominated by a two-sited sequence
of natural numbers extending Carlson’s approach, and we apply this theory
exploiting the Budak-Isik-Pym representation to obtain a partition theory for
the set of rational numbers.
In Chapter 3 we prove recurrence results for topological dynamical systems
indexed by words. In this way we extend the classical theory developed by
Birkhoff, Furstenberg and Weiss of dynamical systems indexed by the natural
numbers to systems indexed by words.
In Chapter 4 we introduce the notion of a rational dynamical system and we
prove (multiple) recurrence results for such systems. We also give some
applications of these topological recurrence results to topology, to
combinatorics, to diophantine approximations and to number theory.
Finally, in Chapter 5, we introduce the notion of a partitionable system over
an infinite semigroup and we state and prove a strong partition theorem for the
semigroup corresponding to each system. We also get partition theorems for
semigroups with digital representation.

Ramsey theory, Partition theorems, Semigroups, Topological dynamical systems, Rational numbers

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https://pergamos.lib.uoa.gr/uoa/dl/object/1308961