|B類(講究) C類(実習)/Category B Category C
|Seminar on Probability Theory 1
Seminar on Probability Theory 2
Seminar on Probability Theory 3
Seminar on Probability Theory 4
Practical Class on Probability Theory 1
Practical Class on Probability Theory 2
Practical Class on Probability Theory 3
Practical Class on Probability Theory 4
|JAERISCH Johannes Klaus Bernhard
|JAERISCH Johannes Klaus Bernhard
|Title: Ergodic theory and fractal geometry for conformal dynamical systems
Dynamical systems provide mathematical models to analyse the time dependence of systems in basically all sciences and our daily life. In this seminar, we study chaotic conformal dynamical systems from the view point of ergodic theory and with applications to fractal geometry.
The aim is to enable the students to perform research in the area of ergodic theory and dynamcal systems with applications to fractal geometry. We provide the necessary guidance so that the students can enter this research area and develop their own ideas. A further aim is to enable the students to improve their ability to communicate in English.
|The students will develop deeper understanding and knowledge of mathematics especially related to ergodic theory and fractal geometry for conformal dynamical systems. Through study and research in this area, the students will develop their learning and problem-solving skills. Further, the students will acquire the capability to contribute to the research on topics of this course. The students will improve their communication skills as well as written and oral presentation skills by giving regular reports on their study and research progress. A thesis is written in the second half of the course.
|A dynamical systems is given by a self-map on a (state) space. Chaos refers to the sensitive dependence of orbits on the initial value. For chaotic systems it is difficult to predict and to analyse the individual behavior of orbit points. The same applies to large particle systems. Ergodic theory aims to analyse the average long-term behavior of such systems from the viewpoint of probability measures. We focus in particular on the so-called thermodynamic formalism originating from statistical physics, which allows us to determine meaningful measures for a given dynamical system.
Chaotic behavior of dynamical systems is often related to highly irregular sets (such as attractors of conformal iterated functions systems, limit sets of Kleinian groups, or Julia sets of complex polynomials) which are called fractal sets. In this seminar, we study the interplay of ergodic theory, dynamical systems and fractal geometry for conformal dynamical systems.
The seminar is used to discuss research problems and ideas how to deal with them. Moreover, the seminar helps to monitor progress and timeline of the research project. For the main part of this course, the students are expected to take the responsibility to organise their research activities, and to monitor and manage their time.
|Background in analysis, measure theory, probability theory, functional analysis is required. Knowledge of ergodic theory is desirable.
|Lectures on probability and measure (in particular, Lebesgue integration and measures on topological spaces), and functional analysis (spectral theory for bounded linear operators).
|Grading is based on seminar performance (attendance, presentation, discussion) and Master's thesis.
|Recommended books will be introduced on an individual basis.
|R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975.
F. Dal’ Bo. Geodesic and horocyclic trajectories: Universitext. Springer-Verlag London, Ltd.,London; EDP Sciences, Les Ulis, 2011.
K. Falconer. Mathematical foundations and applications. Third edition. John Wiley and Sons,Ltd., Chichester, 2014
A. Katok, B. Hasselblatt. A first course in dynamics. With a panorama of recent developments. Cambridge University Press, New York, 2003.
D. Mauldin, M. Urbanski. Graph directed Markov systems. Geometry and dynamics of limitsets. Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003.
Y. Pesin. Dimension theory in dynamical systems. Contemporary views and applications.Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997.
F. Przytycki, M. Urbanski. Conformal fractals: ergodic theory methods. London MathematicalSociety Lecture Note Series, 371. Cambridge University Press, Cambridge, 2010.
P. Walters. An introduction to ergodic theory. Graduate Texts in Mathematics, 79. SpringerVerlag, New York-Berlin, 1982.
|To prepare carefully for seminar presentation and discussion.
|This seminar is in English.
|Please send me an email.
|Every student with strong motivation to explore mathematics is welcome.
|Fractals, Hausdorff dimension, Dynamical systems, Ergodic theory, Thermodynamic formalism
|We meet in classroom if the pandemic situation is acceptable. Otherwise, we meet online using zoom/skype etc., and communicate by email.