Dimension theory for skew products

• Applicant: Dr. Johannes Klaus Bernhard Jaerisch
• Subject Area: Mathematics
• Term: from 2011 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 209502447

• Final Report Year: 2015

Final Report Abstract

This was a successful project investigating new classes of complex systems focusing on their dynamical long-term behaviour and their fractal-geometric aspects. The Julia set of a polynomial map acting on the Riemann sphere is a prominent example of a fractal object which reflects the chaoticity of the long-term behaviour of the iteration of the given polynomial. More precisely, the iteration of the polynomial exhibits a sensitive dependence on the initial point precisely for the points in the Julia set. In modern physics, biology and chemistry as well as in financial mathematics and medicine, fractals provide models of objects in applied sciences. One of the aims is to describe the geometry of Julia sets by means of the underlying dynamical system. In this project we focused on systems which have an infinite generator system. That is, instead of iterating a single polynomial on the Riemann sphere, we consider at each step a polynomial from an infinite collection of polynomials. Regarding real world phenomena it is very natural to allow for many different choices at each step since there are many random terms in nature which may cause different developments. Our main result in this context is a formula for the dimension (in the sense of Hausdorff) of the Julia set of a new class of infinitely generated systems. Regarding the random iteration of polynomial maps on the Riemann sphere, we have also investigated the phenomenon of noise-induced order. This phenomenon means that a random perturbation (or a random choice) stabilises the long-term behaviour of the system. That is, the Julia set disappears with respect to a certain random process. In this case, we study how the probability to experience a certain long-term behaviour depends on the initial condition. In fact, another main result of this project shows that, this probability function shows a very irregular behaviour which can again be investigated by means of fractal geometry. The irregularity can be seen as a gradation between chaos and order.

Publications

• Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method, preprint 2013, 32 pages
  J. Jaerisch and H. Sumi
  
• Multifractal formalism for expanding rational semigroups and random complex dynamical systems, preprint 2013, 22 pages
  J. Jaerisch and H. Sumi
  
• Fractal models for normal subgroups of Schottky groups, Trans. Amer. Math. Soc. 366 (2014), 5453–5485
  J. Jaerisch
  
• Group-extended Markov systems, amenability, and the Perron-Frobenius operator, Proc. Amer. Math. Soc. 143 (2014), 289–300
  J. Jaerisch
  
• Recurrence and pressure for group extensions, Ergodic Theory and Dynamical Systems, online first August 2014
  J. Jaerisch
  
• A lower bound for the exponent of convergence of normal subgroups of Kleinian groups, J. Geom. Anal. 23 (2015), 289–305
  J. Jaerisch