Bers and Hénon, Painlevé and Schrödinger

Cantat, Serge

doi:10.1215/00127094-2009-042

Cited by 63 publications

(165 citation statements)

References 51 publications

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“…This is the first part of a series of two papers (see [6]), the aim of which is to describe the dynamics of a polynomial action of the group preserves a holomorphic geometric structure, and to apply this classification to provide a galoisian proof of the irreducibility of the sixth Painlevé equation.…”

Section: Introductionmentioning

confidence: 99%

Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

Cantat¹,

Loray²

2009

Ann. inst. Fourier

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Section: Introductionmentioning

confidence: 99%

Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

Cantat¹,

Loray²

2009

Ann. inst. Fourier

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“…This paper concerns the case when S is nonorientable and Φ is nontrivial. When S is nonorientable, it is homeomorphic to either a two-holed cross-surface C (2,0) or a one-holed Klein bottle C (1,1) . (See, for example, Norbury [23] for a lucid description of these surfaces.)…”

Section: Contentsmentioning

confidence: 99%

Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane

et al. 2019

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The automorphisms of a two-generator free group F 2 acting on the space of orientation-preserving isometric actions of F 2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group Γ on R 3 by polynomial automorphisms preserving the cubic polynomialand an area form on the level surfaces κ −1 Φ (k). The Fricke space of marked hyperbolic structures on the 2-holed projective plane with funnels or cusps identifies with the subsetThe generalized Fricke space of marked hyperbolic structures on the 1-holed Klein bottle with a funnel, a cusp, or a conical singularity identifies with the subset F (C 1,1 ) ⊂ R 3 defined byWe show that Γ acts properly on the subsets Γ • F(C 0,2 ) and Γ • F (C 1,1 ). Furthermore for each k < 2, the action of Γ is ergodic on the complement of Γ • F(C 0,2 ) in κ −1 Φ (k) for k < −14. In particular, the action is ergodic on all of κ −1 Φ (k) for −14 ≤ k < 2. For k > 2, the orbit Γ • F(C 1,1 ) is open and dense in κ −1 Φ (k). We conjecture its complement has measure zero.

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“…However, in many cases (such as when G is a complex Lie group) the boundary of this open set admits a chaotic Mod(S)-action (Souto Storm [51], Goldman [19], Goldman-McShane-Stantchev-Tan [23], and Maloni-Palesi [39]. In an important special case, Cantat [10] proves the existence of orbits (when G = SL(2, C) and S is a punctured torus) whose closure contain both SU(2)-characters and characters of discrete embeddings. This uses work of Cantat-Loray [11] which also relates character varieties to the sixth Painléve equation.…”

Section: Noncompact Groupsmentioning

confidence: 99%

Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

Forni¹,

Goldman²

2018

Geometry and Physics: Volume II

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We extend Teichmüller dynamics to a flow on the total space of a flat bundle of deformation spaces Rep(π, G) of representations of the fundamental group π of a fixed surface S. The resulting dynamical system is a continuous version of the action of the mapping class group Mod(S) of S on Rep(π, G). We observe how ergodic properties of the Mod(S)-action relate to this flow. When G is compact, this flow is strongly mixing over each component of Rep(π, G) and of each stratum of the Teichmüller unit sphere bundle over the Riemann moduli space M(S). We prove ergodicity for the analogous lift of the Weil-Petersson geodesic local. flow.Date: August 1, 2017. 2010 Mathematics Subject Classification. Primary: 58D27 Moduli problems for differential geometric structures; Secondary: 37A99 Ergodic Theory, 57M50 Geometric structures on low-dimensional manifolds, 22F50, Groups as automorphisms of other structures.

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Bers and Hénon, Painlevé and Schrödinger

Cited by 63 publications

References 51 publications

Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane

Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

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