By Jussi Väisälä
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Extra resources for Lectures on n-Dimensional Quasiconformal Mappings
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Let φ : J (f ) → (0, ∞) be a tame potential with a H¨ older exponent β. Then we have the following. (a) The number 1 is a simple isolated eigenvalue of the operator Lˆφ : Hβ → Hβ and all other eigenvalues are contained in a disk of radius strictly smaller than 1. 4, we have Lˆφ = Q1 + S, where Q1 : Hβ → Cρφ is a projector on the eigenspace Cρφ (given by the formula Q1 (g) = ( g dmφ )ρφ , Q1 ◦ S = S ◦ Q1 = 0 and ||S n ||β ≤ Cξ n for some constant C > 0, some constant ξ ∈ (0, 1) and all n ≥ 1. Here is a useful application.
2. For every loosely tame potential φ = −t log |f |τ + h : J (f ) → C, h ∈ Hw β , there is cβ > 0 such that |Sn φ(fv−n (y)) − Sn φ(fv−n (x))| ≤ cβ Vβ (h)|y − x|β 23 24 5. PERRON–FROBENIUS OPERATORS AND GENERALIZED CONFORMAL MEASURES for all n ≥ 0, all z ∈ J (f ), all v ∈ f −n (z) and all x, y ∈ D(z, δ). 7. We also have a dynamically H¨older property for loosely tame potentials. 3. If φ : J (f ) → C is a (t, β)–loosely tame potential, then there exists c = cφ > 0 depending only on β and Vβ (φ) such that exp Sn φ(fv−n (y)) − exp Sn φ(fv−n (x)) ≤ c exp Sn φ(fv−n (x)) |y − x|β for all n ≥ 0, all z ∈ J (f ), all v ∈ f −n (z) and all x, y ∈ D(z, δ).
11 is complete. 4. Thermodynamical formalism We can now establish the following main result of this chapter. 15. If f : C → C function, then for every tame potential φ the following are true. (1) The topological pressure P(φ) = limn→∞ n1 log Lnφ (w) exists and is independent of w ∈ J (f ). (2) There exists a unique ρe−φ -conformal measure mφ and necessarily ρ = eP(φ) . e. µφ is f -invariant and equivalent to mφ . (3) Both measures mφ and µφ are ergodic and supported on the radial (or conical) Julia set Jr (f ).