Thermodynamic Formalism, Dynamics and Dimensions

University Of North Texas, Denton TX

Investigators

Abstract

In this project the principal investigator proposes to further advance and develop the methods for investigations of dynamical, statistical and geometrical aspects of Hilbertian hyperbolic discrete groups, random dynamical systems, one-dimensional lattice gasses, geometry, and dynamics of meromorphic transcendental functions, holomorphic endomorphisms of compact complex manifolds, and continuity of Hausdorff measures for conformal iterated function systems and conformal expanding repellers. Aided by concepts and techniques of dynamical systems, ergodic theory, statistical physics, functional analysis, geometric measure theory, complex analysis, probability theory, and algebraic and differential geometry, appropriate forms of thermodynamic formalism, both deterministic and random, for those systems will be constructed and investigated. The project will involve the analysis of transfer operators, Gibbs and equilibrium states, Julia sets of meromorphic functions, and limit sets of Hilbertian discrete groups, as well as Hausdorff measures and dimensions of attractors of graph-directed Markov systems. The fact that the concepts, techniques and methods of the project, while dynamical in essence, are nevertheless created through the interplay of the branches of mathematics and physics indicated above, will have interesting consequences. The project will shed light on these fields themselves, may stimulate the development of techniques and methods in these areas, and in particular, may cause their growth in response to demands coming from the theory of dynamical systems. Along these lines, the project assumes cooperation of the principal investigator with several specialists in those fields. Such joint work is expected to broaden their mutual professional expertise and should give rise to enhancement of the investigated domains. The active involvement of graduate students is an integral part of the proposed work. The students are expected to gradually master the topics of the proposed research, to learn more about geometric measure theory, the theory of transcendental meromorphic and entire functions, algebraic geometry and other subjects, and finally to contribute to the project their own creative work. The proposed research is expected to result in advanced graduate courses and to attract to Denton scholars who by delivering colloquium and seminar lectures will interact with and scientifically stimulate graduate students and faculty in Denton.