Average measure theoretic entropy for a family of expanding on average random Blaschke products

TL;DR

This work develops a computable formula for the average metric (measure-theoretic) entropy of expanding-on-average random Blaschke product cocycles, extending the deterministic results of Pujals–Roberts–Shub to the random setting. It constructs a random absolutely continuous invariant measure with densities $h_\omega$ and uses the transfer operator to express the entropy as $h_\mu(\mathcal{T}) = h_{\mathbb{P}}(\sigma) + \int_{\Omega} \int_{\mathbb{T}} \log|T'_\omega| \, d\mu_\omega \, d\mathbb{P}(\omega)$, tying fibre entropy to the average derivative growth. The main advance is a computable formula for the average entropy over a 1-parameter family of cocycles: $\bar{h}(\mathcal{T}) = \int_{\Omega} \int_{\mathbb{T}} \log|T'_\omega| \, dm \, d\mathbb{P}(\omega) + h_{\mathbb{P}}(\sigma)$, showing that averaging over rotations captures the mean expansion on the circle. The paper substantiates the theory with concrete examples, including a two-map Blaschke cocycle, and discusses the sufficiency (but not necessity) of admissibility conditions for the results to hold, highlighting the robustness and limitations of the approach.

Abstract

This work gives a computable formula for the average measure theoretic entropy of a family of expanding on average random Blaschke products, generalizing work by Pujals, Roberts and Shub [Expanding maps of the circle revisited: positive Lyapunov exponents in a rich family. $\textit{Ergodic Theory Dynam. Systems.}$ $\textbf{26}(6)$ $(2006),$ $1931$-$1937$] to the random setting. In doing so, we describe the random invariant measure and associated measure theoretic entropy for a class of admissible random Blaschke products, allowing for maps which are not necessarily expanding and may even have an attracting fixed point.

Figures (2)

• Figure 1: $t$ vs $S(t)=(Q^{-1}TQ)(t)$ where $T(z)=-\left(\frac{z-0.4}{1-0.4z}\right)^2, Q(t)=e^{2\pi i t}.$
• Figure 2: The plot shows $t$ vs $h_{\mu_{\theta}}^{fib}(\mathcal{T}_{\theta}),$ where $\theta=e^{2\pi i t},$ for $(\mathcal{T}_{\theta},\sigma_{1})$ and $(\mathcal{T}_{\theta}, \sigma_{2}).$

Theorems & Definitions (48)

• Definition 1.1: Transfer operator
• Theorem 1.2
• Theorem 1.3: Average metric entropy of an admissible Blaschke product cocycle
• Definition 2.1: Inner function
• Definition 2.2: Finite Blaschke product
• Remark 2.3
• Theorem 2.4: expandPujals
• Proposition 2.5: topics_complex
• Lemma 2.6
• Theorem 2.7