By Volker Mayer
The thermodynamical formalism has been constructed by way of the authors for a truly normal type of transcendental meromorphic features. A functionality of this classification is named dynamically (semi) general. the foremost element within the authors' prior paper (2008) was once that one labored with a good selected Riemannian metric house and that the Nevanlinna concept used to be hired. within the current manuscript the authors first enhance upon their past paper in offering a scientific account of the thermodynamical formalism for the sort of meromorphic functionality f and all potentials which are Holder perturbations of -t log /f'/omega. during this basic atmosphere, they turn out the variational precept, they express the life and specialty of Gibbs states (with the definition properly tailored for the transcendental case) and equilibrium states of such potentials, they usually display that they coincide. there's additionally given an in depth description of spectral and asymptotic houses (spectral hole, Ionescu-Tulcea and Marinescu Inequality) of Perron-Frobenius operators, and their stochastic results akin to the important restrict Theorem, K-mixing, and exponential decay of correlations
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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 ).