Random dynamical systems 1000-1M24LUD
The purpose of the lecture is to introduce the basic themes of random dynamical systems. We do not assume any knowledge of the general theory of dynamical systems. We assume that the student has taken a course in Probability II.
Scope:
One-dimensional random real and complex dynamical systems (including the action of random circle homeomorphisms, random maps of the interval, Mobius transformations, iterated nonhyperbolic function systems). This material will fill the initial part of the lecture. It requires no prior preparation. However, it can be a very good source of problems (easier and more involved ) to solve on your own; in particular- good topics for master thesis.
Invariant measures for random systems and stationary measures for processes generated by random systems
Random subsets of R^n
Random measures on Polish spaces, topologies in the space of random measures, compactness
Strange (i.e. different than expected) phenomena in random dynamics
Random fractal sets and their application
Random dynamical systems depending on a parameter.
Type of course
Mode
Prerequisites (description)
Course coordinators
Learning outcomes
The participant in this course will be introduced to the basic tools of random dynamic systems and new threads of research. The course can be a starting point for one's own research work or for the preparation of a master thesis.
Assessment criteria
Regular attendance and activity in class.
Preparation and presentation of a talk on the proposed topic.
Preparing an essay based on a a selected item (article) from the proposed additional literature.
Bibliography
BIBLIOGRAPHY:
BOOKS (fragments)
H. Crauel, Random probability measures on Polish spaces - available on Research gate,
L. Arnold, Random dynamical systems, Springer, 1998
A. Navas, Groups of circle diffeomorphisms, Chicago Lectures in Mathematic Series, 2011; similar text of the same author available on Arxiv
We shall also (partially) work with recent research papers which appeared recently. Below is a list of papers to choose.
A.Bonifant, J. Milnor, Schwarzian derivatives and cylinder maps, Holomorphic dyna-mics and renormalization, 1–21, Fields Inst. Commun., 53, Amer. Math. Soc., Providence, RI, 2008
L. Alseda, M.Misiurewicz, Skew product attractors and concavity, Proc Amer.
Math.Soc.143 (2015), 703-716 L. Alseda, M.Misiurewicz, Random interval homeomorphisms, Publ. Mat. (2014), 15–36 Proceedings of New Trends in Dynamical Systems. Salou, 2012.
A. Ambroladze, H. Wallin, Random iteration of Mobius transformations and Furstenberg’s theorem, Ergodic Theory Dynam. Systems 20 (2000), 953-962.
T.Szarek, A. Zdunik, Stability of Iterated Function Systems on the circle, Bull London Math. Soc. (48) 2016
K.Lech, A.Zdunik, Total disconnectedness of Julia sets of random
quadratic polynomials. Ergodic Theory Dynam. Systems 42 (2022), no. 5, 1764–1780.
H. H. Rugh: On the dimensions of conformal repellers. Randomness and parameter dependency, Ann Math. 168 (2008)
V. Mayer, M. Urbański, A. Zdunik: Real analyticity for random dynamics of transcen- ental functions, Ergodic Theory Dynam. Systems 40(2020), 490–520
D. Belayev, S. Smirnov: Random conformal snowflakes Ann. Math. 172 (2010),
and, possibly other papers which we shall find interesting to discuss.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: