On the Lyapunov spectrum for rational maps

Abstract

We study the dimension spectrum for Lyapunov exponents for rational maps on the Riemann sphere.

Lyapunov spectrum for exceptional rational maps

“…The result of the above theorem has been shown in [7] in the particular case that f is non-exceptional.…”

“…In particular, continuing the investigations in [7], we are interested in the case that the map f is exceptional. Slightly modifying [10, Section 1.3], we call f exceptional if there exists a finite, nonempty, and forward invariant set Σ ′ ⊂ J such that…”

“…To prove our main result, in this paper we will create new technical tools in order to deal with exceptional rational maps and then show how these tools can be applied to adapt the original proofs in [7]. The paper is organized as follows.…”

“…Finally, in Section 5 we apply these tools to prove Theorem 1. In Section 5.1 we provide a lower bound for dimension using the fact that for any rational map we can find an increasing family of uniformly expanding Cantor repellers contained in J using a construction of bridges that has been established in [7] and applies to the setting of this paper without changes. In Section 5.2 we provide an upper bound for dimension applying Frostman's Lemma to an appropriate σ-finite conformal measure at a conical point.…”

“…By Propositions 2 and 4 this pressure coincides with P (φ t ) and is obtained by taking a supremum over uniformly expanding repellers. Note that, given t ∈ R and ε > 0 and a uniformly expanding repeller Λ, by [7,Lemma 3] there exists a positive integer n and an f n -ECR Λ ′ ⊂ Λ such that…”