Teichmueller flow and Weil-Petersson flow

Abstract

For an oriented surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2, let Q(S) and Q W P (S) be the moduli space of area one quadratic differentials and of quadratic differentials of unit norm for the Weil-Petersson metric, respectively. We show that there is a Borel subset E of Q(S) which is invariant under the Teichmüller flow Φ t T and of full measure for every invariant Borel probability measure, and there is a measurable map Λ : E → Q W P (S) which conjugates Φ t T |E into the Weil-Petersson flow Φ t W P . This conjugacy induces a continuous injection of the space of all Φ t T -invariant Borel probability measures on Q(S) into the space of all Φ t W P -invariant Borel probability measures on Q W P (S). The map Θ is not surjective, but its image contains the Lebesgue Liouville measure. A measure not in the image corresponds to a locally finite infinite invariant Borel measure on Q(S).

Rates of mixing for the Weil–Petersson geodesic flow: exponential mixing in exceptional moduli Spaces

“…For 3g − 3 + n > 1, the WP flow is not Anosov, even when viewed in the Teichmüller metric (or an equivalent Riemannian metric such as in [15]), but it might be fruitful to study the flow from this perspective. We remark here that Hamenstädt [11] has recently constructed measurable orbit equivalences between the WP and Teichmüller geodesic flows for all 3g − 3 + n ≥ 1.…”

“…m can be further decomposed, as follows. First, using (14) to bound |u n |, the assumption that ∇K = O(δ −3 ) together with (11) to bound ∇ JVn K , and the fact from Lemma 2.6 that (j m − j n ) = j n O(t n /t), we have that…”

“…), which uses the bounds on ∇K and ∇ 2 K , as well as (11) and Lemma 2.4. The details are left to the patient reader.…”