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.

Project Report

Nineteen research articles have been written, seven of them already published, seven accepted for publication, and five of them submitted for publication in peer reviewed scientific journals. They will extend the theory of dynamical systems, ergodic theory, geometric measure theory, number theory, and probability theory. The PI plans to incorporate the findings in his subsequent research articles and books. The same applies to his students and collaborators and other researchers working in the above fields. Three doctoral students were provided opportunities to do research in the field of dynamical systems and have graduated with doctoral degrees. They will further teach other students in their institutions based on what they learned while working the project. The nineteen research papers produced while working on the project will enhance the teaching abilities of the PI, his students and other scholars working in the field of dynamical systems and ergodic theory. Many students and faculty members at the Mathematics Department of the University of North Texas have been exposed to lectures of the experts in the field with international reputation. The results obtained provided dimension estimates of limit sets of iterated function systems including those with overlaps. Stochastic and fractal properties of equilibrium measures of holomorphic endomorphisms of the Riemann sphere and higher dimensional complex projective spaces, with respect to Holder continuous potentials with pressure gap, were established using the concept of fine inducing introduced in these papers. .Motivated by statistical physics, the thermodynamic formalism for a modified shift map has been built and its key ergodic properties have been established. A strongest version of geometric rigidity for transcendental meromorphic functions has been proved. Finer stochastic properties, such as the Law of Iterated Logarithm, of dynamical systems generated by transcendental meromorphic functions have been obtained. With a heave use of functional analytic methods, In three research papers, continuity properties of numerical value of Hausdorff measure for conformal systems exhibiting sufficient features of hyperbolicity, have been established. It concerned in particular conformal expanding repellers, countable alphabet iterated function system, and parabolic rational functions. In several papers Diophantine properties of those measures that are invariant under some conformal like dynamical systems, have been proved. Two papers concerned non-autonomous dynamical systems. Appropriate versions of Bowen’s formula for the Hausdorff dimension of the limit sets have been established, and their regularity properties have been investigated. In particular, examples showing the break down of real-analyticity, even differentiability, have been constructed. The concept of transversality for finitely generated semigroups of rational functions have been introduced, and an appropriate dimension formula, holding for almost all parameters, has been proved.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Bruce P. Palka
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University of North Texas
United States
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