Doctoral Dissertation uoadl:3221773 165 Read counter

Department of Mathematics

Library of the School of Science

Library of the School of Science

2022-06-30

2022

Karageorgos Dimitrios

Μιχαήλ Ανούσης, Καθηγητής, Πανεπιστήμιο Αιγαίου

Απόστολος Γιαννόπουλος, Καθηγητής, ΕΚΠΑ

Κωνσταντίνος Γρυλλάκης, Αναπληρωτής Καθηγητής, ΕΚΠΑ

Παντελής Δοδός, Αναπληρωτής Καθηγητής, ΕΚΠΑ

Κωνσταντίνος Τύρος, Αναπληρωτής Καθηγητής, ΕΚΠΑ

Βασιλική Φαρμάκη, Καθηγήτρια, ΕΚΠΑ

Νίκος Φραντζικινάκης, Καθηγητής, Πανεπιστήμιο Κρήτης

Απόστολος Γιαννόπουλος, Καθηγητής, ΕΚΠΑ

Κωνσταντίνος Γρυλλάκης, Αναπληρωτής Καθηγητής, ΕΚΠΑ

Παντελής Δοδός, Αναπληρωτής Καθηγητής, ΕΚΠΑ

Κωνσταντίνος Τύρος, Αναπληρωτής Καθηγητής, ΕΚΠΑ

Βασιλική Φαρμάκη, Καθηγήτρια, ΕΚΠΑ

Νίκος Φραντζικινάκης, Καθηγητής, Πανεπιστήμιο Κρήτης

Θέματα Εργοδικής Θεωρίας Ramsey

Greek

Themes of Ergodic Ramsey Rheory

This dissertation consists of three parts. In the first part we introduce the class of coideals on an infinite directed set X, and becomes possible to prove partition results for the ordered finite or infinite sequences in X with respect to a given coideal on X. Our theory extends the classical topological Ramsey theory, and in addition includes as particular cases the corresponding theory for coideals on the set of natural numbers proved by Louveau, Mathias, Farah and Todorcevic, the Milliken-Taylor partition theorems for sequences of finite subsets of natural numbers, and the partition theorems for sequences of words proved by Carlson, Bergelson-Blass-Hindman, Farmaki.

In the second part we prove recurrence and multiple recurrence results for topological dynamical systems indexed by an arbitrary directed partial semigroup with respect to a coideal basis suitable for this semigroup, but otherwise arbitrary. Our theory includes recurrence and multiple recurrence results for topological dynamical systems, indexed by natural numbers, or by finite non-empty subsets of natural numbers, proved by Furstenberg and Weiss, and analogous results for topological dynamical systems indexed by words.

In the third part we deal with the convergence of ergodic means and the consequences they have on recurrence problems, in combinatorics and number theory. Exploiting the equidistribution properties of polynomial sequences and following the methods developed by Leibman and Frantzikinakis, we show that the ergodic averages with iterates given by the integer part of strongly independent real valued polynomials converge in mean to the expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications. Furthermore we get the respective expected limits and combinatorial results for multiple averages for a single sequence, as well as for several sequences along prime numbers.

In the second part we prove recurrence and multiple recurrence results for topological dynamical systems indexed by an arbitrary directed partial semigroup with respect to a coideal basis suitable for this semigroup, but otherwise arbitrary. Our theory includes recurrence and multiple recurrence results for topological dynamical systems, indexed by natural numbers, or by finite non-empty subsets of natural numbers, proved by Furstenberg and Weiss, and analogous results for topological dynamical systems indexed by words.

In the third part we deal with the convergence of ergodic means and the consequences they have on recurrence problems, in combinatorics and number theory. Exploiting the equidistribution properties of polynomial sequences and following the methods developed by Leibman and Frantzikinakis, we show that the ergodic averages with iterates given by the integer part of strongly independent real valued polynomials converge in mean to the expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications. Furthermore we get the respective expected limits and combinatorial results for multiple averages for a single sequence, as well as for several sequences along prime numbers.

Science

Topological Ramsey Theory, Coideals, Topological Dynamical Systems, Ergocic Theory, Equidistribution, Prime Numbers

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