Dynamical Days in Jizzakh: An International School on Dynamical Systems

Standard Pairs, Projective Cones, and Limit Theorems

September 2026
Jizzakh State Pedagogical University

Schedule

Starting: 14 September 2026
End: 18 September 2026
Six lectures of 1:00 each.

The plan of the course is as follows:
  1. Standard pairs and coupling for smooth one-dimensional maps. Exercises on coupling
  2. Cones and Hilbert metric
  3. Large deviations
  4. Central Limit Theorem and Linear response.

Here are some notes about dynamical systems (for the students' eyes only) that I hope may help. In particular, the appendices contain all the functional analysis needed to follow the course.

  • An idiosyncratic introduction to Dynamical Systems. These are notes (in a draft form, read at your own risk) intended for people with little or no knowledge of dynamical systems; the relevant part for this course starts from chapter 7. Moreover, the Appendices A, C, D, F, and H may be useful.
  • Fast slow systems. This article discusses convergence to a Brownian motion.

    More information on transfer operators can be found in the following books:

  • Transfer operators in Hyperbolic Dynamics An introduction, C.Liverani, M. Demers and N. Kiamari. 33 Colloquio Brasilero de Matematica. Editora do IMPA. pp.252 (2021)
  • V. Baladi. Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics: Volume 16 (2000) https://doi.org/10.1142/3657
  • Boyarsky, Abraham; Gò ra, Pawel Laws of chaos. Invariant measures and dynamical systems in one dimension. Probab. Appl.Birkhauser Boston, Inc., Boston, MA, (1997).

    The first is partially a review, so some topics are not treated in full detail. The second is much more detailed, although it covers less material. The third is a bit dated and covers only one-dimensional maps, but it contains some nice facts.

    Here are some articles that contain material relevant to the course: section 1.7 of these notes describes the use of standard pairs for multidimensional piecewise continuous maps.

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