Entropy Theory. Technion. Spring 2020

Thursdays 10:30-12:30.

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


It is planned to convey working knowledge of (dynamical) entropy theory.


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.

  1. Einsiedler, Lindenstrauss, Ward - Entropy In Ergodic Theory And Homogeneous Dynamics link
  2. Ledrappier - Information And Entropy In Dynamical Systems: An Introduction link
  3. Ledrappier - Positivity of the exponent for stationary sequences of matrices link
  4. and (Springer link)
  5. Viana - Lectures on Lyapunov Exponents
  6. Einsiedler, Lindenstrauss - Diagonal actions on locally homogeneous spaces (Pisa Notes) link
  7. Aaronson - Introduction to Infinite Ergodic Theory
  8. Aaronson - Course Notes on Measure Theory. Tel Aviv University
  9. Viana, Oliveira - Foundations of Ergodic Theory
  10. Einsiedler, Ward - Ergodic Theory with a view towards Number Theory
  11. Matilla - Geometry of Sets and Measures in Euclidean Spaces
  12. Rokhlin - Lectures on the entropy theory of measure-preserving transformations link
  13. Adler, Weiss - Entropy, A Complete Metric Invariant For Automorphisms Of The Torus link
  14. Khayutin - Large Deviations and Effective Equidistribution link
  15. Kadyrov - Effective uniqueness of Parry measure and exceptional sets in ergodic theory link
  16. Rühr - Effectivity of Uniqueness of the Maximal Entropy Measure on p-adic homogeneous spaces link
  17. David, Shapira - Equidistribution of divergent orbits and continued fraction expansion of rationals link
  18. Einsiedler, Lindenstrauss, Michel, Venkatesh - The distribution of closed geodesics on the modular surface, and Duke's theorem link
  19. Wieser - Linnik’s Problems And Maximal Entropy Methods link