Entry requirements

Measure and Integration Theory I

Learning outcomes

In order to pass the course (grade 3) the student should be able to

• use the concept measure preservative mapping;
• use the concept ergodicity;
• give an account of the elementary ergodic theorems;
• use the concepts mixing and weak mixing;
• give an account of the concepts induced mapping, invariant measure, ergodic division, and measure rigidity;
• give an account of the applications of ergodic theory within dynamical systems, number theory and geometry.

Content

Justifications of the study of ergodic theory within dynamical systems, number theory and geometry. Measure preservative mappings. Recurrence. Ergodicity. Unitary operators. Ergodic theorems: in mean and pointwise. Mixing and weak mixing. Induced mappings. Applications within number theory: continued fractions. The Gaussian measure. Existence of invariant measures. Ergodic partitions. Uniqe ergodicity. Rigidity of measures. Equal distribution.

Instruction

Lectures and problem solving sessions.

Assessment

Written Moreover, assignments during the course in accordance with instructions at course start.