Statistical Properties of Dynamical Systems

Statistical Properties of Dynamical Systems

Spring Semester 2026
Department of Mathematics
University of Maryland

Schedule

Starting: 26 January, 2026.
End: 08 May 2026.

Monday 13:00-14:15, Room MTH B0429.
Wednesday 13:00-14:15, Room MTH B0429.

INFOS:

Syllabus, and References

Many dynamical systems behave similarly to a stochastic process; they are (very loosely) called "chaotic".
The course aims to illustrate several different techniques for studying and making precise such phenomena.
I particular, I will discuss the following properties:

  • invariant measures and decay of correlations
  • statistical stability and linear response
  • Central limit Theorem, Large deviations, Invariance principle, extreme values statistics ...

    Such concepts will be investigated by analyzing classes of systems of increasing complexity. More precisely:

  • Expanding maps (smooth and non-smooth)
  • Uniformly hyperbolic maps
  • Uniformly hyperbolic flows
  • Partially hyperbolic maps
  • non-uniformly hyperbolic systems

    I will try to provide some notes for the material that cannot be easily found in print. Some of the topics can be found in:

  • 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.

    Notes

    Here are some notes (for the students eyes only) that I hope may help. In particular, the appendices contain all the functional analysis needed to follow the course.
    I will updated them continuously, without warning, so check for the current version.
    They are most likely full of mistakes, so read at your own risk.
    I will try to upload more as we prooceed (but do not keep your hopes too high).

  • An idiosyncratic introduction to Dynamical Systems. These are notes intended for people with little or no knowledge of synamical systems. However, staring from chaper 6 they are increasingly relevant also for the present course. Moreover, the Appendices A, C, D, F, G will be rather important for us.
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