Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Δημήτριος Γατζούρας, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ (επιβλέπων)
Παντελεήμων Δοδός-Ντοντός, Αν. Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Κωνσταντίνος Τύρος, Αν. Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Εργοδικά Θεωρήματα
Translated title:
Ergodic Theorems
Summary:
The main purpose of this thesis is the study of ergodic theorems mainly in
the context of functional analysis. We also present some ergodic theorems
in the classical setting of ergodic theory, i.e. measure preserving systems.
In particular, we are concerned about the convergence of averages of
powers of a bounded linear operator on a Banach space. In detail:
• In Chapter 1 we are interested in the case where the averages arise
from a measure preserving transformation. We prove the ergodic
theorems of Birkhoff and Von Neumann and we give the motivation
to study more general averages. At the end of the chapter, generalizing
Birkhoff's theorem we prove the subadditive ergodic theorem of Kingman
and one of its applications regarding the convergence of random matrices.
• In Chapter 2 we study the pointwise and uniform convergence of averages
arising from a bounded linear operator in a Banach space. The theorem of
Eberlein is of great importance since it provides necessary and sufficient
condition of the convergence of such averages. The study of ergodic
theorems in this general context leads us to a spectral theorem
concerning the powers of a quasi compact operator.
• In Chapter 3 we study positive contractions in L1. We prove Hopf's decomposition
and Chacon - Ornestein' s theorem which assure us that the pointwise limit of
the ratio of averages studied in Chapter 2 exists almost everywhere on a particular
set. Brunel's lemma allows us to describe the limit of the ratio, a result originally
due to Neveu-Chacon.
Main subject category:
Science
Keywords:
Ergodic theory, Functional analysis, measure preserving system, convergence of averages, positive contractions
