Rates of recurrence for real extensions of complex dynamics

Abstract

In this note, we consider the family of hyperbolic quadratic polynomials Pc (z)=z 2 + c ; c ε M, the Mandelbrot set and study ideas on the distribution of values of periodic orbits to the corresponding hyperbolic Julia sets and derive asymptotic results. Here, we alternate between symbolic dynamics and the hyperbolic polynomial map mentioned above. Later, we prove the same results for pairs of points in the Julia set. These results complement the central limit type results due to Lalley. Our proof requires application of ideas from thermodynamic formalism.