Summer School in Dynamics at ICTP - Course material

Week 2 - Ergodicity in smooth dynamics (Jana Rodriguez-Hertz, Amie Wilkinson, Khadim War)

• Lecture 1: Exercises from Amie, Exercises from Jana.
• Lecture 2: Exercises from Amie, Exercises from Khadim, the Puzzle from Amie.
• Lecture 3: Exercises from Amie, Exercises from Jana.
• Lecture 4: Lecture Notes and Exercises from Amie, Exercises from Khadim.
• Lecture 5: Exercises from Jana, Lecture Notes and Exercises from Amie.
  Reference from Amie:
  • Fubini Foiled: Katok's Paradoxical Example in Measure Theory by John Milnor, The Mathematical Intelligencer (1997) 19: 30.

Week 2 - Renormalization in entropy zero systems (Corinna Ulcigrai)

• Lecture 1: Exercises, Slides.
  References:
  • The first three pages of The Geometry of Markoff Numbers by Caroline Series, Math. Intelligencer 7, 1985, 20-29.
  • Lecture notes by Sébastien Ferenczi about complexity and solutions to some of the exercises.
• Lecture 2: Exercises, Slides.
  References:
  • Again follows The Geometry of Markoff Numbers by Caroline Series.
  • The chapter Sturmian Sequences by Pierre Arnoux, which is Chapter 6 in Substitutions in Dynamics by N. Pytheas Fogg, Arithmetics and Combinatorics, volume 1794 of Lecture Notes in Mathematics, Springer-Verlag, Berlin (2002), pp. 143-195.
    It can be downloaded here.
• Lecture 3: Slides, Exercises.
  References:
  • More detailed Slides from past seminars about characterisation of cutting sequences in the square and the octagon.
  • Symbolic coding for linear trajectories in the regular octagon by John Smillie and Corinna Ulcigrai.
  • Flat Surfaces by Anton Zorich (see below).
  • Cutting sequences on Bouw-Möller surfaces: an S-adic characterization by Diana Davis, Irene Pasquinelli and Corinna Ulcigrai
    (the introduction is easy to read and contains a summary of the characterisation for square and octagon as well as the results of the paper).
  • Lecture notes on Ergodic theory of translation surfaces by Howard Masur.
• Lecture 4: Slides, Exercises.
  References:
  • Continued fraction algorithms for interval exchange maps: an introduction by Jean Christophe Yoccoz, Frontiers in number theory, physics, and geometry. I, 401-435, Springer, Berlin, 2006.
  • Lecture notes by Marcelo Viana.
  • Lecture notes by Jean Christophe Yoccoz from the Clay Summer School in Pisa.
• Lecture 5: Exercises.

Week 1 - Rotations of the circle and renormalization (Corinna Ulcigrai)

• Lecture 1: Lecture notes, Exercises, Solutions.
• Lecture 2: Lecture notes, Exercises, Solutions.
• Lecture 3: Notes, Exercises.
  References:
  • Chapter 10 of the book Topics in ergodic theory by Yakov G. Sinai (Princeton Mathematical Series, Volume 44 (1994) ISBN: 0-691-03277-7),
  • Pages 5-6 of the Lecture Notes by Davoud Cheraghi (see Additional References below), but careful to the different notations,
  • Section 5.4 of Flat Surfaces by Anton Zorich, Frontiers in number theory, physics, and geometry. I, 437-583, Springer, Berlin, 2006 .
• Lecture 4: Exercises.
  References:
  • Part (page1 and page2) of A Limit Theorem for Birkoff sums of non-integrable functions over rotations by Yakov G. Sinai and Corinna Ulcigrai, Geometric and probabilistic structures in dynamics, 317-340, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008.
• Lecture 5: Lecture notes, Exercises.

Week 1 - Piecewise expanding maps of the circle (Jana Rodriguez-Hertz)

• Lecture 1: Slides, Exercises.
• Lecture 2: Slides, Exercises.
• Lecture 3: Slides, Exercises.
• Lecture 4: Slides, Exercises.
• Lecture 5: Slides, Exercises.

Week 1 - Introduction to ergodic theory (Oliver Butterley, Irene Pasquinelli, Stefano Luzzatto, Lucia Simonelli, Davide Ravotti)

• Lecture 1: Slides, Lecture notes: Metric spaces and Measure spaces, Exercises.
• Lecture 2: Lecture notes, Exercises.
• Lecture 3: Lecture notes, Notes with Exercises and Solutions.
• Lecture 4: Lecture notes, Exercises.
• Lecture 5: Lecture notes, Exercises.

Additional references

• Lecture notes in Dynamical Systems and Ergodic Theory by Corinna Ulcigrai:
  Part I (first examples of dynamical systems and their properties: rotations, doubling map, Gauss map and continued fractions, symbolic coding, ...);
  Part II (shift spaces and topological dynamics: topologically transitive, topologically mixing, entropy, ...);
  Part III (ergodic theory: measure preserving, Poincaré recurrence, ergodicity, Birkhoff ergodic theorem, ...).
• An introductory course by Davoud Cheraghi in dynamics which has good overlap with the topics of the first week courses
  like rotations (Weyl theorem, renormalization, ...), expanding maps (degree, symbolic coding, ...) can be found here.
• More advanced lecture notes on Ergodic Theory by Michael Hochman can be found here.
• Lecture notes by Stefano Luzzatto: an Introduction to Dynamical System and an Introduction to Smooth Ergodic Theory