In this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely P c (z) = z 2 + c with |c| < 1 and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely P (a 1 ,a 0 ) (z) = z 3 + a 1 z + a 0 with |a i | < 1, restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtain results pertaining to the dependence of the behaviour of these exponents on the parameters describing the polynomial map. We achieve this using the theory of thermodynamic formalism, the pressure function in particular.

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, R 1 , R 2 , • • • , R N , each with degree d i ; 1 ≤ i ≤ N and with at least one R i in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanskii (equilibrium) measure associated to some real-valued Hölder continuous function defined on the Julia set of the skewproduct map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

No abstract