EIDIKA THEMATA II (V.16): Ergodikī THeōría

Course Information
TitleΕΙΔΙΚΑ ΘΕΜΑΤΑ ΙΙ (Β.16): Εργοδική Θεωρία / EIDIKA THEMATA II (V.16): Ergodikī THeōría
Code0871
FacultySciences
SchoolMathematics
Cycle / Level2nd / Postgraduate
Teaching PeriodWinter/Spring
CommonNo
StatusActive
Course ID600022466

Programme of Study: PMS Tmīmatos Mathīmatikṓn (2018-sīmera)

Registered students: 7
OrientationAttendance TypeSemesterYearECTS
THEŌRĪTIKA MATHĪMATIKACore Courses2110
Class Information
Academic Year2022 – 2023
Class PeriodSpring
Faculty Instructors
Class ID
600227800
Course Type 2021
Specific Foundation
Mode of Delivery
  • Face to face
  • Distance learning
Digital Course Content
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
  • Greek (Instruction, Examination)
  • English (Instruction, Examination)
Prerequisites
General Prerequisites
General knowledge in Mathematical Analysis.
Learning Outcomes
Upon successful completion of the course, students will be able to study the phenomenon of recurrence (for points and sets) in systems, which will be able to decide whether they are ergodic, mixing, or weakly mixing. They will also be able to study the equidistribution of real sequences, through the Weyl and van der Corput criteria.
General Competences
  • Retrieve, analyse and synthesise data and information, with the use of necessary technologies
  • Adapt to new situations
  • Make decisions
  • Work autonomously
  • Generate new research ideas
  • Be critical and self-critical
  • Advance free, creative and causative thinking
Course Content (Syllabus)
The goal of the course is to study the following topics: • Measure preserving systems • Recurrence and ergodicity • Mean ergodic theorem • Equidistributed sequences • van der Corput's trick • Polynomial recurrence • Mixing and weak mixing systems • Compact functions • Multiple recurrence in weakly mixing systems • Multiple recurrence in compact systems • Complementarity between compactness and weak mixing • Reduction of recurrence to ergodic systems • Decomposition of measure into ergodic components • Furstenberg's correspondence principle • Szemerédi's theorem Bibliography: • M. Einsiedler and T. Ward. Ergodic theory with a view towards number theory, volume 259 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 2011. • H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. • P. Walters. An Introduction to Ergodic Theory, Springer-Verlag, 1982.
Keywords
Recurrence, weak mixing, compact functions, compact systems, ergodicity, arithmetic progressions, density
Educational Material Types
  • Notes
  • Slide presentations
  • Multimedia
  • Book
Use of Information and Communication Technologies
Use of ICT
  • Use of ICT in Course Teaching
  • Use of ICT in Communication with Students
  • Use of ICT in Student Assessment
Description
Use of computer program MATLAB for examples, exercises and research in and outside of the classroom. Online hours for exercises.
Course Organization
ActivitiesWorkloadECTSIndividualTeamworkErasmus
Lectures391.3
Reading Assigment1836.1
Tutorial130.4
Project200.7
Written assigments401.3
Exams50.2
Total30010
Student Assessment
Description
Homework during the semester and final written exam.
Student Assessment methods
  • Written Exam with Short Answer Questions (Formative, Summative)
  • Written Exam with Extended Answer Questions (Formative, Summative)
  • Written Assignment (Formative, Summative)
  • Performance / Staging (Formative, Summative)
  • Written Exam with Problem Solving (Formative, Summative)
Bibliography
Additional bibliography for study
• M. Einsiedler and T. Ward. Ergodic theory with a view towards number theory, volume 259 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 2011. • H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. • P. Walters. An Introduction to Ergodic Theory, Springer-Verlag, 1982.
Last Update
09-05-2025