Statistics of multipliers for hyperbolic rational maps

In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.

[1]  J. Milnor.  Dynamics in one complex variable , 2000 .

[2]  H. Oh,et al.  Prime number theorems and holonomies for hyperbolic rational maps , 2016, 1603.00107.

[3]  David Ruelle,et al.  The thermodynamic formalism for expanding maps , 1989 .

[4]  Fr'ed'eric Naud,et al.  Expanding maps on Cantor sets and analytic continuation of zeta functions , 2005 .

[5]  Alan F. Beardon,et al.  Iteration of Rational Functions , 1991 .

[6]  M. Pollicott,et al.  Exponential error terms for growth functions on negatively curved surfaces , 1998 .

[7]  David Ruelle,et al.  An extension of the theory of Fredholm determinants , 1990 .

[8]  Norbert Steinmetz.  Rational iteration : complex analytic dynamical systems , 1993 .

[9]  H. Krieger,et al.  Complex dynamics , 2012, Veterinary Record.

[10]  D. Sullivan.  Conformal dynamical systems , 1983 .

[11]  David Ruelle,et al.  Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics , 2004 .

[12]  M. Pollicott,et al.  Correlations of Length Spectra for Negatively Curved Manifolds , 2013 .

[13]  DISTRIBUTION OF ERGODIC SUMS FOR HYPERBOLIC MAPS , 2004 .

[14]  H. Weiss,et al.  A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions , 1997 .

[15]  M. Lyubich.  Entropy of analytic endomorphisms of the riemannian sphere , 1981 .

[16]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[17]  Correlations for pairs of closed geodesics , 2006 .

[18]  D. Dolgopyat.  On decay of correlations in Anosov flows , 1998 .

[19]  A. Eremenko,et al.  Rational maps with real multipliers , 2011 .