The principal investigators propose to further advance and develop the theory and methods for investigations of conformal and holomorphic dynamical systems. This includes deterministic and random iterations of transcendental entire and meromorphic functions, graph directed Markov systems on Hilbert spaces and dimension spectra for finite-dimensional iterated function systems. In conjunction with the methods of algebraic geometry and differential geometry, the thermodynamic formalism for rational functions on complex projective spaces is proposed to be built. Furthermore, the theory of rational semigroups is proposed to be advanced. The methods laid out are also used to investigate divergent ergodic averages along squares and higher powers, and the structure of homeomorphic Bernoulli trial measures on the Cantor space is analyzed. Apart from ergodic theory and dynamical systems, the proposed research also involves such areas of mathematics as complex analysis, probability theory, geometric measure theory, topology, dimension theory, number theory, algebraic geometry and differential geometry.
