Thermodynamics, dimension and the Weil–Petersson metric

Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

“…Moreover, there exists a family of conjugating homeomorphisms π λ : ∂D → ∂D on the boundary satisfying γ λ • π λ = π λ • γ 0 for corresponding elements γ 0 ∈ 0 and γ λ ∈ λ . In [10], McMullen gave an interesting interpretation of · WP in terms of the Hausdorff dimension dim H (π λ (l)) of the image π λ (l) of Lebesgue measure l on ∂D. …”

The geometry of the Weil-Petersson metric in complex dynamics

“…Since v is an infinitesimal version of the Schwarzian derivative, it is naturally a quadratic differential. In [McM2], McMullen observed that…”

“…one can define ||ḟ t || 2 WP by the integral average (1.4), while for z → z d , one can use a more complicated integral average described in [McM2].…”

“…Recall from [McM2] that a continuous function h : D → C is almost-invariant if for any > 0, there exists r( ) < 1, so that for any orbit z → f (z) → · · · → f •n (z) contained in {z : 1 − r ≤ |z| < 1}, we have |h(z) − h(f •n (z))| < . The argument above also tells us that: Theorem 7.3.…”

“…In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed in [McM2]. We show that the Weil-Petersson metric is incomplete and study its metric completion.…”