Introduction to Smooth Ergodic Theory
SISSA 2021
Smooth Ergodic Theory is the study of dynamical systems on smooth manifolds from a probabilistic and statistical perspective.
In this course we will focus on some relatively simple systems which are highly “chaotic” (unpredictable) and show that they are nevertheless very “regular” (predictable) from a statistical point of view. This can be observed in many reallife dynamical systems, from simple cointossing to the weather, where short term outcomes are unpredictable but long term averages are very stable. The underlying philosophical purpose of the course is to try to understand the mechanisms which allow these two apparently contradictory features to coexist.
More technically, the course will begin with a survey of some basic dynamical systems (contraction maps, circle rotations, full branch maps, symbolic systems,..), then introduce various concepts (invariant measures, ergodic measures, physical measures...), results (Poincarè Recurrence, Birkhoff’s Ergodic Theorem, and techniques (distortion calculations, push forward of measures,...) through which their statistical properties can be understood.
It is a first introductory course on the subject and there are no particular prerequisites except for standard elementary notions in topology and measure theory.
In this course we will focus on some relatively simple systems which are highly “chaotic” (unpredictable) and show that they are nevertheless very “regular” (predictable) from a statistical point of view. This can be observed in many reallife dynamical systems, from simple cointossing to the weather, where short term outcomes are unpredictable but long term averages are very stable. The underlying philosophical purpose of the course is to try to understand the mechanisms which allow these two apparently contradictory features to coexist.
More technically, the course will begin with a survey of some basic dynamical systems (contraction maps, circle rotations, full branch maps, symbolic systems,..), then introduce various concepts (invariant measures, ergodic measures, physical measures...), results (Poincarè Recurrence, Birkhoff’s Ergodic Theorem, and techniques (distortion calculations, push forward of measures,...) through which their statistical properties can be understood.
It is a first introductory course on the subject and there are no particular prerequisites except for standard elementary notions in topology and measure theory.
Classes will be in persons and the schedule available on the SISSA website
Lecture notes with exercises are available HERE.
The lecture notes are divided into three parts. Part I introduces the main definitions and examples of Dynamical Systems. Part II is an introduction to a more systematic study of Dynamical Systems from a topological point of view, with a particular focus on the concept of topological conjugacy which is an equivalence relation on the space of all Dynamical Systems. Part III introduces the main ideas and results of Ergodic Theory in some particularly simple settings.
The course covers the following chapters of the notes:
Part I  Fundamental Notions  Chapters 14
Part II  Main Examples  Section 5.3 (Full Branch Maps)
Part IV  Differentiable Ergodic Theory  Chapters 1621 (up to 21.2 and not including 21.3)
The programme includes all proofs and exercises in the sections mentioned above, except for the proof of Birkhoff's Theorems in Chapters 16 and 17.
Classes will be recorded and videos or links to the videos posted here.
www.dropbox.com/sh/fjgvxe8azygyyq9/AABQpcOfOTRenzpfrTjBZJxBa?dl=0 or see embedded videos below.
The lecture notes are divided into three parts. Part I introduces the main definitions and examples of Dynamical Systems. Part II is an introduction to a more systematic study of Dynamical Systems from a topological point of view, with a particular focus on the concept of topological conjugacy which is an equivalence relation on the space of all Dynamical Systems. Part III introduces the main ideas and results of Ergodic Theory in some particularly simple settings.
The course covers the following chapters of the notes:
Part I  Fundamental Notions  Chapters 14
Part II  Main Examples  Section 5.3 (Full Branch Maps)
Part IV  Differentiable Ergodic Theory  Chapters 1621 (up to 21.2 and not including 21.3)
The programme includes all proofs and exercises in the sections mentioned above, except for the proof of Birkhoff's Theorems in Chapters 16 and 17.
Classes will be recorded and videos or links to the videos posted here.
www.dropbox.com/sh/fjgvxe8azygyyq9/AABQpcOfOTRenzpfrTjBZJxBa?dl=0 or see embedded videos below.
Lecture 1 Part 1  Orbits

Lecture 1 Part 2  Limit Sets



Lecture 2 Part 1  Time Averages

Lecture 2 Part 2  Physical Measures



Lecture 3 Part 1  Push Forward of Measures

Lecture 3 Part 2  Invariant Measures

Lecture 4 Part 1  Ergodic Measures as Extremal

Lecture 4 Part 2  Birkhoff's Ergodic Theorem

If you are following this course in person or online please fill in the following short registration form.