Note: the starting times (note change!!) are modified as following:

MONDAYs: Lecture start at 8.15 (with a 10 minutes break 9.00-9.10)

TUESDAYs: Lecture start at 10.20 (with a 10 minutes break 10.05-11.15)

Recordings are NOT Recordings are NOT available for this class.   Only the first class is recorded (link above). The lecture notes (available under the Downloads Tab) cover quite systematically the lecture content. A list of topics covered and references to the notes is available here below.

List of Topics covered (with references to Lecture notes Sections):

• Lecture 1 (Monday, 16/2): Introduction and overview (Section 1.1)
• Lecture 2 (Tuesday, 17.2): The logistic family; graphical analysis, attracting and repelling fixed points (Section 1.2, with all subsections)
• Lecture 3 (Monday 23/2) Rotations of the circle: dichotomy for rotations and Weyl equidistribution (Section 1.3)
• Lecture 4 (Tuesday, 24/2) The doubling map: periodic points, shift map and (semi)-conjugacies (beginning of Section 1.5, Sec 1.5.1, Sect. 1.5.2 until Prop.1).
• Lecture 5 (Monday 2/3 The doubling map continued: itineraries and coding (Sec 1.5.2, existence of dense orbits (completed Sec 1.5), definition of baker map (beginning of Sec. 1.7)
• Lecture 6 (Tuesday 3/3) Itineraries and conjugaces for the baker map (completed Sec. 1.7), definition of the CAT map (beginning Sec. 1.8)
• Lecture 7 (Monday 9/3) The CAT map and hyperbolic toral automorphisms: definition of hyperbolicity and results on periodic points (completed Section 1.8)
• Lecture 8 (Tuesday 10/3) Gauss map and continued fractions (Section 1.9), definition of topological dynamical systems and examples of metric spaces (beginning of Sect 2.1)
• Lecture 9 (Monday 16/3) First topological properties: transitivity, minimality, topological mixing (Sec 2.1.1. without example of baker map)
• Lecture 10 (Tuesday 17/3) top. mixing for the baker map (Prop. 2 in Sect 2.1.1), Topological conjugacies (Sec. 2.2.1), properties and examples (Extra: Minkowski question mark function); Definition of SDIC and expansivity (Sec. 2.2.2),
• Lecture 11 (Monday 23/3) examples of expansive (doubling map), and SDIC but not expansive (CAT map) and definition of Devanay chaotic (Sec 2.2.2); introduction to topological entropy (Sec. 2.3); definition of separated sets (beginning of Sec 2.3.1);
• Lecture 12 (Tuesday 24/3) definition of topological entropy via seprated sets (Sec 2.3.1) and equivalence of definition via spanning sets (Sec 2.3.2); computation of entropy of the doubling map and of the rotation;
• Lecture 13 (Monday 30/3): Entropy of the CAT map building separating and spanning sets (Sec. 2.4.1); definition entropy via covers (beginnig of Sec 2.4.2);
• Lecture 14 (Tuesday 31/3): Equivalence of definition of entropy of via covers (Sec 2.4.2); Symbolic coding, shift spaces (Sec 2.5) and definition of a topological Markov chain (Def. 2.5.3 in Sec. 2.5);
• Lecture 15 (Tuesday 14/4): Graph associated to a Markov chain (Sec. 2.5 continued) and number of paths Lemma (Lemma 2.5.2 in Sec. 2.5); Distance, balls and cylinders for a shift space (beginning of Sec 2.6); definition of irreducibility and aperiodicity/primitivity and condition of topological transitivity and topological mixing (end of Sec 2.6) ;
• Lecture 16 (Monday 20/4): Repetition of proof of topological mixing (Proof of (2) of Thm 2.6.1 in Sec 2.6); Topological entropy of Markov chains via Covering sets (sketch, last subsection of Sec 2.6), Symbolic coding of the CAT map: building a Markov partition (Sec 2.7)
• Lecture 17 (Tuesday 21/4, by Yuriy Tumarkin): Measure preserving transformations (Sec 3.2)
• Lecture 18 (Monday 27/4) Criterium for measure preserving via integrals. Poincare Recurrence (returns and strong form, with infinitely many returns). Paradox? Kac's Lemma statement (correction tomorrow).  
• Lecture 19 (Tuesday 28/4) Comments/extras on last class (variations of measure invariant via integrals, multiple recurrence and use of induced map). Definition of tower/skyscrapers presentation and proof of Kac's lemma (Section 3.5). Statement of Krylov-Bogolyubov Thm (on existence of invariant measures) and preliminaries (Sec.).
• Lecture 20 (Monday 4/5) Proof of Krylov-Bogolyubov Thm (Sec 3.4). Definition of ergodicity, ergodicity of the doubling map (from the definition using Lebesgue density, Ex. 3.6.2), example of non ergodicity (rational rotation Ex. 3.6.2) (Sec 3.5);
• Lecture 21 (Tuesday 5/5) Ergodicity via invariant functions (Lemma 3.6.1) (proof), properties of Fourier series (beginning Sec 3.7), Proofs of ergodicity via Fourier series (irrational rotation and doubling map); Boltzmann ergodic hypothesis;
• Lecture 22 (Monday 12/5) Ergodicity of toral automorphisms (via Fourier series); Statement of Birkhoff ergodic theorem (version for ergodic and for measure preserving T); Applications of B.E.T. (frequencies of visits, Borel numbers, CF entries distribution);
• Lecture 23 (Tuesday 13/5) Proof of Gauss-Kuzmin CF distribution and geometric mean of CF entries; von Neumann mean ergodic theorem (statement and proof); definition of Bernoulli measures on full shifts;
• Lecture 24 (Monday 18/5 part 1) Shift-invariance of Bernoulli measures; definition and invariance of Markov measures. Definition of mixing; mixing of the doubling map; mixing of Markov chains assuming aperiodicity;
• Lecture 25 (Monday 18/5 part 2) Mixing of Bernoulli shift; Statement of ergodic theorem for Markov chains (ETMC) and Perron-Frobenius (PF); Google PageRank algorithm; Sketch of proofs (PF and ETMC) using contractions of the Hilbert metrix on the symplex.
• Lecture 26 (Tue 19/5) Mixing implies ergodicity; relation between mixing and topological mixing; Equivalent characterizations of ergodicity (via B.E.T. and 'mixing in average'). Unique ergodicity (definition) and Oxtoby ergodic thm. Definition of flow.
• Lecture 27 (Tue 26/5) Linear flow on the torus and other examples of flows. Modification of definitions of topological and ergodic properies for continuous time dynamical systems. Ergodicity of irrational linear flows using Poincare sections. A toy model of Anosov flow obtained suspending the CAT map. (all in Chapter 4) Poincare-Hopt method to prove ergodicity (to be added to Chapter 4 in the next few days).