Bers, Brown and Lyapunov Romain Dujardin ´ Ecole Polytechnique April 14 th , 2012 Romain Dujardin Bers, Brown and Lyapunov
Plan 0. Prologue : bifurcation currents for rational maps. 1. Stability/bifurcation dichotomy for Kleinian groups. 2. Lyapunov exponent of a surface group representation 3. The degree of a projective structure. Joint work (in progress) with Bertrand Deroin (Orsay) Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Let ( f λ ) λ ∈ Λ be a holomorphic family of rational mappings of degree d on P 1 . Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Let ( f λ ) λ ∈ Λ be a holomorphic family of rational mappings of degree d on P 1 . For every λ , f λ admits a natural invariant probability measure µ λ (Brolin-Lyubich measure). We can consider the Lyapunov exponent function. � � � � ( f λ ) ′ ( z ) � d µ λ ( z ) λ �− → χ ( f λ ) = log Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Let ( f λ ) λ ∈ Λ be a holomorphic family of rational mappings of degree d on P 1 . For every λ , f λ admits a natural invariant probability measure µ λ (Brolin-Lyubich measure). We can consider the Lyapunov exponent function. � � � � ( f λ ) ′ ( z ) � d µ λ ( z ) λ �− → χ ( f λ ) = log It is continuous and plurisubharmonic ∗ (psh) on Λ. Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Let ( f λ ) λ ∈ Λ be a holomorphic family of rational mappings of degree d on P 1 . For every λ , f λ admits a natural invariant probability measure µ λ (Brolin-Lyubich measure). We can consider the Lyapunov exponent function. � � � � ( f λ ) ′ ( z ) � d µ λ ( z ) λ �− → χ ( f λ ) = log It is continuous and plurisubharmonic ∗ (psh) on Λ. Definition (DeMarco) The bifurcation current is T bif = dd c λ ( χ ( f λ )) Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Results. Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Results. ◮ Support theorem (DeMarco) Supp ( T bif ) is the bifurcation locus of the family. Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Results. ◮ Support theorem (DeMarco) Supp ( T bif ) is the bifurcation locus of the family. ◮ Equidistribution of special subvarieties. (D.-Favre, Bassanelli-Berteloot) Natural sequences of hypersurfaces associated with bifurcations equidistribute towards T bif . Romain Dujardin Bers, Brown and Lyapunov
Prologue : bifurcation currents for rational maps. Results. ◮ Support theorem (DeMarco) Supp ( T bif ) is the bifurcation locus of the family. ◮ Equidistribution of special subvarieties. (D.-Favre, Bassanelli-Berteloot) Natural sequences of hypersurfaces associated with bifurcations equidistribute towards T bif . ◮ Formulas for Lyapunov exponent. Example : the Manning-Przytycki formula : if f λ is a monic polynomial of degree d then � χ ( f λ ) = log d + G f λ ( c ) . c critical Romain Dujardin Bers, Brown and Lyapunov
Sullivan’s dictionary Aim : translate these concepts into the context of families of subgroups of Aut ( P 1 ) = PSL (2 , C ). Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups � a b � ◮ Aut ( P 1 ) ≃ PSL (2 , C ) via az + b cz + d ↔ . Fix �·� on SL (2 , C ). c d Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups � a b � ◮ Aut ( P 1 ) ≃ PSL (2 , C ) via az + b cz + d ↔ . Fix �·� on SL (2 , C ). c d obius transformation γ ( z ) = az + b ◮ A M¨ cz + d � = id has a type model : z �→ e i θ z tr 2 ( γ ) ∈ [0 , 4) elliptic tr 2 ( γ ) = 4 parabolic model : z �→ z + 1 tr 2 ( γ ) / loxodromic model : z �→ kz ∈ [0 , 4] Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups Let G be a finitely generated group, Λ a complex manifold, and ρ = ( ρ λ ) λ ∈ Λ : Λ × G → PSL (2 , C ) be a holomorphic family of representations of G (i.e. it is holomorphic in λ and a homomorphism in g ). Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups Let G be a finitely generated group, Λ a complex manifold, and ρ = ( ρ λ ) λ ∈ Λ : Λ × G → PSL (2 , C ) be a holomorphic family of representations of G (i.e. it is holomorphic in λ and a homomorphism in g ). Standing assumptions : (R1) the family is non-trivial (R2) ρ λ 0 is faithful for some λ 0 (R3) for every λ , ρ λ is non-elementary (i.e. does not have a finite orbit on H 3 ∪ P 1 ). Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups Let G be a finitely generated group, Λ a complex manifold, and ρ = ( ρ λ ) λ ∈ Λ : Λ × G → PSL (2 , C ) be a holomorphic family of representations of G (i.e. it is holomorphic in λ and a homomorphism in g ). Standing assumptions : (R1) the family is non-trivial (R2) ρ λ 0 is faithful for some λ 0 (R3) for every λ , ρ λ is non-elementary (i.e. does not have a finite orbit on H 3 ∪ P 1 ). (or sometimes (R3’) there is λ 0 , s.t. ρ λ 0 is non-elementary. ) Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups Theorem (Sullivan, and also Bers, Marden, etc.) Let ( ρ λ ) λ ∈ Λ be as above, and Ω ⊂ Λ be a connected open subset. The following are equivalent : 1. ∀ λ ∈ Ω , ρ λ is discrete ; 2. ∀ λ ∈ Ω , ρ λ is faithful ; 3. for every g ∈ G, ρ λ ( g ) does not change type as λ ranges in Ω ; 4. for all λ , λ ′ ∈ Ω , ρ λ and ρ λ ′ are quasiconformally conjugate on P 1 , i.e. there exists a qc homeo φ : P 1 → P 1 s.t. ∀ g ∈ G, ρ λ 0 ( g ) ◦ φ = φ ◦ ρ λ 1 ( g ) . Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups Theorem (Sullivan, and also Bers, Marden, etc.) Let ( ρ λ ) λ ∈ Λ be as above, and Ω ⊂ Λ be a connected open subset. The following are equivalent : 1. ∀ λ ∈ Ω , ρ λ is discrete ; 2. ∀ λ ∈ Ω , ρ λ is faithful ; 3. for every g ∈ G, ρ λ ( g ) does not change type as λ ranges in Ω ; 4. for all λ , λ ′ ∈ Ω , ρ λ and ρ λ ′ are quasiconformally conjugate on P 1 , i.e. there exists a qc homeo φ : P 1 → P 1 s.t. ∀ g ∈ G, ρ λ 0 ( g ) ◦ φ = φ ◦ ρ λ 1 ( g ) . Such a family is said to be stable on Ω. Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups So we get a decomposition Λ = Stab ∪ Bif as a maximal domain of local stability and its complement. Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups So we get a decomposition Λ = Stab ∪ Bif as a maximal domain of local stability and its complement. We also have identified a dense codimension 1 phenomenon responsible for bifurcations : Corollary For every t ∈ [0 , 4], � { λ, tr 2 ( ρ λ ( g )) = t } ⊃ Bif . g ∈ G Romain Dujardin Bers, Brown and Lyapunov
1. Stability/bifurcation dichotomy for Kleinian groups So we get a decomposition Λ = Stab ∪ Bif as a maximal domain of local stability and its complement. We also have identified a dense codimension 1 phenomenon responsible for bifurcations : Corollary For every t ∈ [0 , 4], � { λ, tr 2 ( ρ λ ( g )) = t } ⊃ Bif . g ∈ G Note : Bif has non-empty interior (Margulis-Zassenhaus lemma) so Stab is not dense in this setting. Romain Dujardin Bers, Brown and Lyapunov
2. Lyapunov exponent of a surface group representation For the remainder of the talk G = π 1 ( X , ⋆ ) is the fundamental group of a compact connected surface of genus ≥ 2, endowed with Riemann surface structure. Endow X with its Poincar´ e metric. Romain Dujardin Bers, Brown and Lyapunov
2. Lyapunov exponent of a surface group representation For the remainder of the talk G = π 1 ( X , ⋆ ) is the fundamental group of a compact connected surface of genus ≥ 2, endowed with Riemann surface structure. Endow X with its Poincar´ e metric. Note : everything should work for (hyperbolic) surfaces of finite type (technically much more difficult) Romain Dujardin Bers, Brown and Lyapunov
2. Lyapunov exponent of a surface group representation Let ρ : G → PSL (2 , C ) be a non-elementary representation. For v a unit tangent vector at ⋆ , let γ ⋆, v the corresponding unit speed half geodesic. For t > 0 close the path γ ⋆, v | [0 , t ] by a path of length ≤ diam( X ) returning to ⋆ . We get a loop � γ t . Romain Dujardin Bers, Brown and Lyapunov
2. Lyapunov exponent of a surface group representation Let ρ : G → PSL (2 , C ) be a non-elementary representation. For v a unit tangent vector at ⋆ , let γ ⋆, v the corresponding unit speed half geodesic. For t > 0 close the path γ ⋆, v | [0 , t ] by a path of length ≤ diam( X ) returning to ⋆ . We get a loop � γ t . Definition-Proposition For a.e. v ∈ T 1 ⋆ X , the limit 1 χ geodesic ( ρ ) = lim t log � ρ ([ � γ t ]) � t →∞ exists and does not depend on v . Romain Dujardin Bers, Brown and Lyapunov
Recommend
More recommend
