Ergodic Theory and Dynamical Systems

Equality of pressures for rational functions

a1 Institute of Mathematics Polish Academy of Sciences, ul. Sniadeckich 8, 00950 Warszawa, Poland (e-mail:
a2 Departamento de Matematica, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile (e-mail:
a3 Department of Mathematics, Royal Institute of Technology, Stockholm 10044, Sweden (e-mail:

Article author query
przytycki f   [Google Scholar] 
rivera-letelier j   [Google Scholar] 
smirnov s   [Google Scholar] 


We prove that for all rational functions f on the Riemann sphere and potential $-t\ln|f'|, t\ge 0$ all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc. 351(5) (1999), 2081–2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet–Eckmann condition. In Appendix A, we discuss the case t < 0.

(Received February 21 2003)
(Revised September 3 2003)