ABSTRACT

This is a U.S.-Polish cooperative research project that will focus on Ergodic theory and geometry of transcendental entire and meromorphic functions. The principal investigators are Dr. Mariusz Urbanski from the University of North Texas, Professor Janina Kotus from Warsaw University of Technology and Professor Anna Zdunik from Warsaw University.

The main objects of investigation in this project are transcendental entire and meromorphic functions. The topological structure of the Julia sets of such maps has recently been intensively investigated. Fractal properties on the level of Hausdorff dimension have also been dealt with. The research to be done in this project goes to the deeper level of Hausdorff and packing measures for non-hyperbolic exponential maps. Real analyticity of the Hausdorff dimension for the hyperbolic tangent family will be investigated. The main tool of the research will consist of the concept of a conformal measure essentially belonging to the arsenal of thermodynamic formalism going beyond the uniformly hyperbolic systems on compact spaces. Various kinds of transfer operators will also be used frequently. The work on the class of non-hyperbolic exponential maps will primarily concern the subset of the Julia set which carriers all the recurrent and chaotic part of the dynamics. It is conjectured that the appropriate Hausdorff measure of this subset is positive and finite whereas the packing measure is infinite. The next step would be to prove the existence of an invariant measure equivalent with the conformal measure, and to explore its ergodic properties. The case when the parameter lambda is equal to 1/e will also be extensively studied as well as the behavior of the Hausdorff dimension when lambda increases to 1/e. Other problems to be dealt with are the multifractal analysis and the maximizing orbit problem in this context. The investigations of transcendental meromorphic functions will be focused on non-recurrent elliptic functions and non-hyperbolic tangent family. In the context of non-recurrent elliptic functions the main goal is to explain the nature of conformal measures, in particular, to determine whether these measures are purely atomic or atomless. The latter case would open the door to an extensive study of ergodic properties of a sigma-finite invariant measure equivalent with this conformal measure. For the hyperbolic tangent family the goal is to show that the Hausdorff dimension of the Julia set depends on the parameter lambda in a real-analytic manner.

The completion of the project will shed a new light on the evolution of chaotic systems with non-compact phase space. It will enhance the knowledge of long-term behavior of such systems. The nature of the Julia sets, the fractals appearing in many popular publications, will be understood further. Its fundamental geometric properties will be investigated.

This project in mathematics research fulfills the program objectives of bringing together leading experts in the U.S. and Central/Eastern Europe to combine complementary efforts and capabilities in areas of strong mutual interest and competence on the basis of equality, reciprocity, and mutuality of benefit.

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