... [29,§6]) together with the maximal ergodic theorem for Uabas in (9) in Section 2.1.3. More about entropy and the high entropy caseA well-known theorem by Ledrappier and Young [19] relates the entropy, the ... κ(x, x)−1. The additional statement in the parenthesis follows similarly, the onlydifference being the use of a slightly different value for θ in both cases, and then taking the intersection ... clear that ˜γ and hence g−1˜γg ∈ Γ satisfy all the conditions of Theorem 5.1.6. Conclusion of the proof of Theorem 1.3In this section, we conclude the derivation of Theorem 1.3, and its corol-lary,...