The Seminar will be held online only via Zoom.
| Date | Topic | Notes |
|---|---|---|
| 19.03.2020 | Entropy of a Partition. Jensen's inequality. Reference: Chapter 1 [1] | Notes |
| 26.03.2020 | Kolmogorov-Sinai Entropy. Sinai's generator Theorem. Reference: Chapter 1 [1] | Notes |
| 02.04.2020 | Bernoulli Shift, Conditional measures (disintegration), Uniform integrability of Conditional Information. Reference: Chapter 2 [1], Chapter 1 [6], Chapter 5 [8] | Notes |
| 16.04.2020 | Shannon-McMillan-Breiman Theorem. Shannon Source Coding Theorem. Reference: Chapter 1,3 [1] | Notes |
| 23.04.2020 | Brin-Katok Theorem, Endomorphism of Tori. Chapter 3 [1], Chapter 4 [2] | Notes |
| 30.04.2020 | Invariance principle after Ledrappier, Furstenberg's theorem on positive top Lyapunov exponent. Reference: [3] | Notes |
| 07.05.2020 | Lyapunov exponents bound entropy after Ledrappier. Dimension of a measure and Besicovitch Covering Theorem. Reference: [3] | Notes. Updated notes (Nov 2020) |
| 14.05.2020 | Zero and Maximal entropy, Abramov-Rokhlin Formula, Convex Combinations, Pinsker and Tail algebra. Reference: Chapter 2 [1] The multiple mixing discussion after the lecture followed Chapter 15 [11] | Notes |
| 21.05.2020 | Fibre measures, Leafwise measures, Entropy contribution. Reference: [5] | Notes |
| 28.05.2020 | Flowers, Subordinate sigma algebras, Relation between Entropy and Entropy contribution of the horospherical subgroup. Reference: [5] | Notes |
| 04.06.2020 | Intrinsically ergodic systems: Invariance on homogeneous spaces ([5]), Unique maximal measure for a Markov shift (called Parry measure) ([12], [1] Chapter 8.1.1). | Notes |
| 11.06.2020 | Large Deviations after Khayutin ([13]) for systems with property (M), Chernoff-Hoeffding Inequality, Pinsker Inequality | Notes |
| 18.06.2020 | Einsiedler inequality: Effective Uniqueness of the maximal entropy measure for Markov shifts and homogeneous spaces ([14], [15]) | Notes |
| 25.06.2020 | Separated sets: Entropy method for equidistribution related to the variational principle. (Pieces of) Divergent orbits of the geodesic flow at rational points on the standard horocycle equidistribute in the space of lattices following David-Shapira ([16]). See also Section 5.3.1 [1] and Section 4 [17]. | Notes |
| 02.07.2020 | Linnik's problems. Equidistribution of integer points on spheres following Wieser ([18]). | Notes |
The course is dedicated to the concept of measure-theoretic information and entropy: We start with a space \(X\) that is partitioned into subsets \(\mathcal{P}=\{P_i\}\), i.e.\ \(X=\bigsqcup_{i=1}^\infty P_i\). Given a probability measure \(\mu\), we have the probabilities \(p_i=\mu(P_i)\) so that \(\sum p_i =1\). The information function is \(I(\mathcal{P})(x)=-\log p_i\) if \(x\in P_i\). The entropy of the partition \(\mathcal{P}\) with respect to \(\mu\) is \[H_\mu(\mathcal{P})=\sum_i-p_i\log{p_i}=\mathbb{E}_\mu[I(\mathcal{P})].\] It provides a mathematical model for the information gained on average by observing to which \(P_i\) a point \(x\in X\) belongs to. An important feature is the following maximal property: If the number of partition elements is fixed of size \(N\) then \(H_\mu(\mathcal{P})\) attains its maximum \(\log N\) if and only if \(p_i=1/N\) for all \(i\). Indeed, this follows from the ``equality case'' of Jensen's inequality.
In ergodic theory, one is given transformation \(T:X\to X\) (moving one unit of time forward) which respects the underlying measure, namely, it preserves integrals. Knowledge about the future trajectory up to time \(n\) of a point is encoded in the partition \( \mathcal{P}_0^n:=\mathcal{P}\vee T^{-1}\mathcal{P}\vee\dots\vee T^{-n}\mathcal{P}=\{\bigcap P_i: P_i\in T^{-i}\mathcal{P}\} \) obtained by refining \(\mathcal{P}\) by its preimages under \(T\). The entropy of the map \(T\) with respect to \(\mathcal{P}\) is the limit \[ h_\mu(T,\mathcal{P})=\lim_{n\to\infty}\frac{1}{n}H_\mu(\mathcal{P}_0^n). \] The Shannon-McMillan-Breiman theorem states that the averaged information function \(\frac{1}{n}I(\mathcal{P}_0^n)\) will converge almost surely to \(h_\mu(T,\mathcal{P})\) if \(\mu\) is ergodic.
We will study these notions in the general context of measure-preserving transformations, and will put considerable attention to the following two systems. The Bernoulli shift \(X=\{x=(\dots,x_{-1},x_0,x_1,\dots): x_i\in\{0,1\}\} \text{ with }(Tx)_i=x_{i+1}\) and the hyperbolic torus automorphism \(X= \mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2 \text{ with } Tx=\left[\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\right]x\).
The concept of conditional expectation is fundamental to the study of entropy theory and we plan to review the required theorems of measure theory. We also introduce the notion of conditional measures: Given a map between probability spaces \(\pi:(X,\mu)\to (Y,\nu)\) we construct measures \(\mu_y\) on \(X\) that allow us to disintegrate \(\mu=\int_Y \mu_y d\nu(y)\).
Towards the end of the course we discuss some applications to smooth and random dynamics in the simplest possible setting, namely the description of measures invariant under toral automorphisms using the concept of stable/unstable foliations, and the asymptotic growths behavior of random matrix products \(A_kA_{k-1}...A_1\). If time permits, we may consider dynamics on more general homogeneous spaces.
We plan to cover materials presented in chapters 1-3 from the new book (1). The discussion on stable/unstable manifolds for the toral case will follow (2), (5). Our presentation of stationary measures and Furstenberg's theorem will follow (3) and a related textbook is (4). Disintegration follows presentation of (6),(7) (like (6) but more detailed) and (8). (10) is a nice source for Besicovitch covering Theorem.