Measures, Dimensions and Dynamics

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The project concerns deterministic and random conformal dynamical systems. This includes the theory of conformal iterated function systems (IFS) founded by the proposers , rational functions of the Riemann sphere and entire and meromorphic functions. We propose to study conformal IFS in a Hilbert space, in particular to examine the transfinite dimensions of the limit sets . Extending the work with A. Zdunik to the dimensions greater than 2, we propose to deal with the Hausdorff dimension of the harmonic measure of the limit set (whose closure is a topological Cantor set) of an IFS. Developing our rigidity investigations we intend to prove the geometrical rigidity of higher dimensional limits sets whose closures are topological disks. We intend to provide necessary and sufficient conditions for conformal measures of regular IFS to satisfy the doubling property and as an application we would present a characterization of all subsets of natural numbers which generate the continued fraction systems with conformal measures satisfying the doubling property. Passing to rational functions we propose to deal with no-recurrent maps (more precisely, the critical points contained in the Julia sets are assumed to be non-recurrent). We would like to work on the proof of equality of the Hausdorff dimension and box dimension of these maps, to study various definitions of pressure and the escape rates. Employing the concept of affine laminations, we propose to develop together with M. Lyubich the thermodynamic formalism of semi-hyperbolic rational functions. Together with J. Kotus and A. Zdunik we propose to explore invariant and geometric (Hausdorff, packing) measures for the members of the exponential family, as well as meromorphic functions fitting into Walters' thermodynamic formalism. We intend to deal with recently emerged topic of quantization dimension relating its theory to the multifractal formalism. Our last sub-project concerns geometric measures of random IFS, in particular concerning the existence of an exactpacking measure function.We propose to continue and develop our ongoing research involving measures of various kinds-Hausdorff, packing, conformal, Gibbs states, invariant measures absolutely continuous with respect to Lebesgue measure, etc., and associated functions such as capacities and pressureand finally associated dimensions. These notions are studied from two interlaced viewpoints. One is that of various dynamical systems and the other is that of geometric measure theory as applied to recursively generated objects. These approaches naturally meld with one another and lead to some interesting mixtures of ideas. We have developed an extensive theory of conformal iterated function systems, the iteration of infinitely many conformal maps, which may be hyperbolic (uniformly contracting) or parabolic (at least one map has an indifferent fixed point) and also have made some applications of it to some well known problems, e.g., continued fractions and Apollonian packings. There are yet some fundamental outstanding problems and several others which arise within the context of applications. We want to extend and develop this theory to cover a more structured iteration of maps-whose governed by directed graphs or substitutions and random iterations and an appropriate multifractal formalisn for these systems. This has many applications not only to the study of rational, entire and meromorphic functions, but also to newly emerging theory of quantization dimension problems arising from statistics and engineering. We also intend to apply our theory to limit sets generated by various cellular automata. This could be very important for compressing visual images.