Complex Dynamics and Fractals

Contact

Lecturer: Prof. Manfred Denker

e-mail: denker@math.psu.edu

Teaching Assistant: David Koslicki

e-mail: koslicki@math.psu.edu

Course Outline

A tentative course outline is as follows:

1. Introduction to Thermodynamic Formalism
2. Rational maps: conformal measures, Bowen-McClusky formula, and Hausdorff dimension
3. Overview of an open problem regarding the Martin Boundaries and Julia sets


Click here for the Mathematica (Labs) webpage.

Reading Material

If you are a participant and you are interested in studying these texts, please contact David Koslicki for further information.

Fractal geometry for dynamical systems is closely related to thermodynamic formalism. For an introduction to thermodynamic formalism see:

1. Introduction to thermodynamic formalism. These lecture notes provide a short introduction to the topic. The material will be discussed in the course. Preliminaries are knowledge of measure theory and dynamical systems. These preliminaries are contained in:

2. Ergodic Theory on Compact Spaces. Lecture notes in Math. Springer 1976 (reprint is available by Springer). Sections 1, 2, 3, 8 and 9 should be studied before the course.

There are several books on complex iteration theory, the second topic of the course (both topics will be combined). The material to prepare for the course can be found for example in:

3. L. Carleson, T.W. Gamelin: Complex Dynamics. Springer 1993. Here Chapter I, sections 1,2,3 and 4; and chapter II, sections 1,2,3,4,5,6 are relevant. You may use any other book on the topic as well.

The course will cover fractal properties of Julia sets. An excellent reference for the material and an updated account of knowledge is:

4. F. Przytycki, M. Urbanski: Conformal fractals: ergodic theory methods . London Mathematical Society Lecture Note Series, 371. Cambridge University Press, Cambridge, 2010.
The course will discuss several of the methods used in the book.

5. M. Denker, B.O. Stratmann: Survey of Conformal Measures (The Patterson Measure: Classics, Variations, and Applications).

6. M. Denker: Lectures on Probability and Dynamics.

Projects

Instead of homework, students are expected to complete one of the following projects. Since these projects vary in difficulty and specificity, it is suggested that you consult with Prof. Denker or David Koslicki regarding your choice.

The complete list of available projects for this course can be found in the following pdf document.

References

1. Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Y. Pesin, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, 1998, ISBN: 0-226-66222-5 ( Russian Translation, Moscow--Izevsk, 2002)

2. Equilibrium States in Ergodic Theory, G. Keller, London Mathematical Society Student Texts, 42, Cambridge University Press 1998, ISBN: 0521 59420 0

3. Rational iteration: complex analytic dynamical systems, N. Steinmetz, deGruyter Studies in Mathematics, 16, 1991, ISBN: 3-11-013765-8

4. Iteration of rational functions: complex analytic dynamical systems, A. Beardon, Springer-Verlag, 1991, ISBN: 0-38-797589-6

5. An introduction to infinite ergodic theory, J. Aaronson, American Mathematical Society, 50, 1997, ISBN: 0-82-180494-4

6. An introduction to chaotic dynamical systems, R. Devaney, Westview Press, 1992, ISBN: 978-0201-554-069