Teichmüller spaces and holomorphic motions

Guiding isotopies and holomorphic motions

“…In contrast to our definition of quiding quasiconformal isotopy the definition in [11] and [25] does not require that µ t vary continuously in the L ∞ topology. The idea of using a guiding quasiconformal isotopy is suggested in [7] where it is used to solve a version of the guiding homeomorphism problem.…”

“…A notion very close to the concept of guiding isotopy is the notion of quasiconformal trivializaiton defined in Definition 2.1 of [11] and Definition 1.1 of [25]. In contrast to our definition of quiding quasiconformal isotopy the definition in [11] and [25] does not require that µ t vary continuously in the L ∞ topology.…”

“…Thus for any new point p ∈ C \ E, we can have a holomorphic motion h : ∆ × (E ∪ {p}) → C extending h. A general version of Lemma 5 is proved as [25,Theorem C] and [20,Theorem 5.5]. To complete the proof, we need the λ-Lemma of Mañé, Sad and Sullivan, [23].…”

Extending holomorphic motions and monodromy

“…It follows that Φ : R × E → C is a holomorphic motion of E. Finally, since E is dense in C, Φ can be extended to a holomorphic motion of C by the λ-lemma (see, for example, Lemma 14.1 in [11]. )…”

Regularity and irregularity of fiber dimensions of non-autonomous dynamical systems

“…Hubbard [Hub76] discovered that this is false for holomorphic motions over higher-dimensional parameter spaces and [JM07] contains a simpler example. Nevertheless, we dispose in the following λ-Lemma due to Mitra [Mit00] and Jiang-Mitra [JM07].…”