The Hardy Class of Geometric Models and the Essential Spectral Radius of Composition Operators

Letφbe an analytic map of the disk into itself (that fixes the origin). Then, it is well known thatφinduces a bounded composition operatorCφon the Hardy spaceH2(D). Also, by a classical result of Kœnigs there is an analytic mapσsuch thatσ∘φ=λσ(λ=φ′(0)≠0), which is one-to-one whenφis one-to-one. Recently, P. Bourdon and J. Shapiro found a lower bound for the essential spectral radius ofCφin terms ofλandh(σ)=sup{p>0: σ∈Hp(D)}, and asked for an exact formula. In the following, we assume thatφis one-to-one. We determine the Hardy class ofσin terms of the image setG=σ(D), and then show that Bourdon and Shapiroʹs lower bound is also an upper bound, hence answering their question in the case of univalent symbols. Moreover, in an earlier paper, Shapiro, Smith, and Stegenga introduced the concept of twisted sectors to estimateh(σ), and we showed [P. Poggi-Corradini,Trans. Amer. Math. Soc.348(6) (1996), 2503–2518] how the dynamics of the mapφand some harmonic measure estimates allow one to completely characterize the caseh(σ)=∞ in terms of twisted sectors. By further analyzing this concept, we produce a new characterization ofh(σ) in terms of the invariant sector-like regions that are contained inG=σ(D). This different approach also yields another proof of Bourdon and Shapiroʹs lower bound mentioned above, in our specific case.