Slope problems in the theory of semigroups of holomorphic self-maps - PowerPoint PPT Presentation

Asymptotic behaviour Theorem (The continuous Denjoy-Wolff theorem) Three possible situations can happen: 1 Every iterate ϕ t is the identity; that is, every trajectory is stationary. 2 There exist ω ∈ R \ { 0 } and an automorphism T of D such that ϕ t ( z ) = T − 1 ( e itω T ( z )) . That is, trajectories are rotations inside D . 3 There exists τ ∈ D (unique and called the Denjoy-Wolff point of the semigroup) such that, for every z ∈ D , lim t → + ∞ ϕ t ( z ) = τ . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 4 / 26

Asymptotic behaviour Theorem (The continuous Denjoy-Wolff theorem) Three possible situations can happen: 1 Every iterate ϕ t is the identity; that is, every trajectory is stationary. 2 There exist ω ∈ R \ { 0 } and an automorphism T of D such that ϕ t ( z ) = T − 1 ( e itω T ( z )) . That is, trajectories are rotations inside D . 3 There exists τ ∈ D (unique and called the Denjoy-Wolff point of the semigroup) such that, for every z ∈ D , lim t → + ∞ ϕ t ( z ) = τ . Semigroups verifying (1), (2) or (3) with τ ∈ D are called elliptic. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 4 / 26

Asymptotic behaviour Theorem (The continuous Denjoy-Wolff theorem) Three possible situations can happen: 1 Every iterate ϕ t is the identity; that is, every trajectory is stationary. 2 There exist ω ∈ R \ { 0 } and an automorphism T of D such that ϕ t ( z ) = T − 1 ( e itω T ( z )) . That is, trajectories are rotations inside D . 3 There exists τ ∈ D (unique and called the Denjoy-Wolff point of the semigroup) such that, for every z ∈ D , lim t → + ∞ ϕ t ( z ) = τ . Semigroups verifying (1), (2) or (3) with τ ∈ D are called elliptic. Semigroups verifying (3) with τ ∈ ∂ D are called non-elliptic. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 4 / 26

Hyperbolic and parabolic semigroups S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Hyperbolic and parabolic semigroups Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Hyperbolic and parabolic semigroups Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . Then, there exists λ ∈ [0 , + ∞ ) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Hyperbolic and parabolic semigroups Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . Then, there exists λ ∈ [0 , + ∞ ), called the spectral value of the semigroup ( ϕ t ) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Hyperbolic and parabolic semigroups Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . Then, there exists λ ∈ [0 , + ∞ ), called the spectral value of the semigroup ( ϕ t ) such that r → 1 ϕ ′ t ( rτ ) = e − λt , lim t ≥ 0 . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Hyperbolic and parabolic semigroups Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . Then, there exists λ ∈ [0 , + ∞ ), called the spectral value of the semigroup ( ϕ t ) such that r → 1 ϕ ′ t ( rτ ) = e − λt , lim t ≥ 0 . Hyperbolic semigroups are those with λ > 0. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Hyperbolic and parabolic semigroups Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . Then, there exists λ ∈ [0 , + ∞ ), called the spectral value of the semigroup ( ϕ t ) such that r → 1 ϕ ′ t ( rτ ) = e − λt , lim t ≥ 0 . Hyperbolic semigroups are those with λ > 0. Parabolic semigroups are those with λ = 0. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 5 / 26

Slopes in the context of non-elliptic semigroups (I) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . The set of slopes of ( ϕ t ) when arriving at τ S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . The set of slopes of ( ϕ t ) when arriving at τ , and denoted as Slope[ ϕ t ( z ) , τ ] S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . The set of slopes of ( ϕ t ) when arriving at τ , and denoted as − π 2 , π � � Slope[ ϕ t ( z ) , τ ] ⊂ 2 S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . The set of slopes of ( ϕ t ) when arriving at τ , and denoted as − π 2 , π � � Slope[ ϕ t ( z ) , τ ] ⊂ 2 is defined as the cluster set at t = + ∞ of the curve t ∈ [0 , + ∞ ) �→ Arg(1 − τϕ t ( z )) . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . The set of slopes of ( ϕ t ) when arriving at τ , and denoted as − π 2 , π � � Slope[ ϕ t ( z ) , τ ] ⊂ 2 is defined as the cluster set at t = + ∞ of the curve t ∈ [0 , + ∞ ) �→ Arg(1 − τϕ t ( z )) . � belongs to Slope[ ϕ t ( z ) , τ ] � − π 2 , π In other words, θ ∈ 2 S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (I) Let ( ϕ t ) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂ D . The set of slopes of ( ϕ t ) when arriving at τ , and denoted as − π 2 , π � � Slope[ ϕ t ( z ) , τ ] ⊂ 2 is defined as the cluster set at t = + ∞ of the curve t ∈ [0 , + ∞ ) �→ Arg(1 − τϕ t ( z )) . � belongs to Slope[ ϕ t ( z ) , τ ] if and only if � − π 2 , π In other words, θ ∈ 2 there exists a sequence ( t n ) ⊂ [0 , + ∞ ) converging to + ∞ such that n →∞ Arg(1 − τϕ t n ( z )) = θ. lim S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 6 / 26

Slopes in the context of non-elliptic semigroups (II) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 7 / 26

Slopes in the context of non-elliptic semigroups (II) � − π 2 , π Slope[ ϕ t ( z ) , τ ] is either a point or a closed subinterval of � . 2 S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 7 / 26

Slopes in the context of non-elliptic semigroups (II) � − π 2 , π Slope[ ϕ t ( z ) , τ ] is either a point or a closed subinterval of � . 2 S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 7 / 26

Slopes in the context of non-elliptic semigroups (III) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, it is important to consider when they can be considered “moderated” S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, it is important to consider when they can be considered “moderated” (in this case, the machinery of angular calculus is available). S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, it is important to consider when they can be considered “moderated” (in this case, the machinery of angular calculus is available). By moderated oscillations S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, it is important to consider when they can be considered “moderated” (in this case, the machinery of angular calculus is available). By moderated oscillations, we mean Slope[ ϕ t ( z ) , τ ] is a closed subinterval of ( − π/ 2 , π/ 2). S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, it is important to consider when they can be considered “moderated” (in this case, the machinery of angular calculus is available). By moderated oscillations, we mean Slope[ ϕ t ( z ) , τ ] is a closed subinterval of ( − π/ 2 , π/ 2). ( ϕ t ( z )) converges non-tangentially to τ S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes in the context of non-elliptic semigroups (III) Concerning oscillations, it is important to consider when they can be considered “moderated” (in this case, the machinery of angular calculus is available). By moderated oscillations, we mean Slope[ ϕ t ( z ) , τ ] is a closed subinterval of ( − π/ 2 , π/ 2). ( ϕ t ( z )) converges non-tangentially to τ if Slope[ ϕ t ( z ) , τ ] is a singleton or a closed subinterval of ( − π/ 2 , π/ 2). S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 8 / 26

Slopes: hyperbolic semigroups S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that Slope[ ϕ t ( z ) , τ ] = { θ ( z ) } . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that Slope[ ϕ t ( z ) , τ ] = { θ ( z ) } . Moreover S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that Slope[ ϕ t ( z ) , τ ] = { θ ( z ) } . Moreover 1 θ : D → R is an harmonic function. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that Slope[ ϕ t ( z ) , τ ] = { θ ( z ) } . Moreover 1 θ : D → R is an harmonic function. 2 θ maps surjectively D onto ( − π/ 2 , π/ 2) . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that Slope[ ϕ t ( z ) , τ ] = { θ ( z ) } . Moreover 1 θ : D → R is an harmonic function. 2 θ maps surjectively D onto ( − π/ 2 , π/ 2) . 3 θ ( z 1 ) = θ ( z 2 ) if and only if there is t ≥ 0 such that either ϕ t ( z 1 ) = z 2 or ϕ t ( z 2 ) = z 1 . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Slopes: hyperbolic semigroups Theorem (Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D . Then. for every z ∈ D , there exists θ ( ϕ t , z ) = θ ( z ) ∈ ( − π/ 2 , π/ 2) such that Slope[ ϕ t ( z ) , τ ] = { θ ( z ) } . Moreover 1 θ : D → R is an harmonic function. 2 θ maps surjectively D onto ( − π/ 2 , π/ 2) . 3 θ ( z 1 ) = θ ( z 2 ) if and only if there is t ≥ 0 such that either ϕ t ( z 1 ) = z 2 or ϕ t ( z 2 ) = z 1 . 4 θ (essentially) determines the hyperbolic semigroup ( ϕ t ) . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

Linear models for hyperbolic semigroups (I) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0 S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: 1 S π/λ is the open strip { x + iy ∈ C : x ∈ (0 , π/λ ) , y ∈ R } . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: 1 S π/λ is the open strip { x + iy ∈ C : x ∈ (0 , π/λ ) , y ∈ R } . 2 h is a univalent function from D into S π/λ S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: 1 S π/λ is the open strip { x + iy ∈ C : x ∈ (0 , π/λ ) , y ∈ R } . 2 h is a univalent function from D into S π/λ (hence h ( D ) is a simply connected domain). S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: 1 S π/λ is the open strip { x + iy ∈ C : x ∈ (0 , π/λ ) , y ∈ R } . 2 h is a univalent function from D into S π/λ (hence h ( D ) is a simply connected domain). 3 h ◦ ϕ t ( z ) = h ( z ) + it , z ∈ D , t ≥ 0 S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: 1 S π/λ is the open strip { x + iy ∈ C : x ∈ (0 , π/λ ) , y ∈ R } . 2 h is a univalent function from D into S π/λ (hence h ( D ) is a simply connected domain). 3 h ◦ ϕ t ( z ) = h ( z ) + it , z ∈ D , t ≥ 0 (hence h ( D ) + it ⊂ h ( D ) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (I) Every hyperbolic semigroup ( ϕ t ) with spectral value λ > 0has an essentially unique model ( S π/λ , h, z �→ z + it ). This means: 1 S π/λ is the open strip { x + iy ∈ C : x ∈ (0 , π/λ ) , y ∈ R } . 2 h is a univalent function from D into S π/λ (hence h ( D ) is a simply connected domain). 3 h ◦ ϕ t ( z ) = h ( z ) + it , z ∈ D , t ≥ 0 (hence h ( D ) + it ⊂ h ( D ) − → h ( D ) is a starlike at infinite domain). S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

Linear models for hyperbolic semigroups (II) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 11 / 26

Linear models for hyperbolic semigroups (II) The idea is to look at geometrical aspects of Ω := h ( D ) to describe the behaviour of the slopes of the trajectories. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 11 / 26

Linear models for hyperbolic semigroups (II) The idea is to look at geometrical aspects of Ω := h ( D ) to describe the behaviour of the slopes of the trajectories. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 11 / 26

Hyperbolic semigroups: models and slopes S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Hyperbolic semigroups: models and slopes Theorem (Bracci, Contreras, DM) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Hyperbolic semigroups: models and slopes Theorem (Bracci, Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D , spectral value λ > 0 and model ( S π/λ , h, z �→ z + it ) . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Hyperbolic semigroups: models and slopes Theorem (Bracci, Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D , spectral value λ > 0 and model ( S π/λ , h, z �→ z + it ) .If ( z n ) ⊂ D is a sequence converging to τ with β = n → + ∞ Arg(1 − τz n ) ∈ ( − π/ 2 , π/ 2) , lim S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Hyperbolic semigroups: models and slopes Theorem (Bracci, Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D , spectral value λ > 0 and model ( S π/λ , h, z �→ z + it ) .If ( z n ) ⊂ D is a sequence converging to τ with β = n → + ∞ Arg(1 − τz n ) ∈ ( − π/ 2 , π/ 2) , lim then n →∞ Re h ( z n ) = β λ + π lim 2 λ. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Hyperbolic semigroups: models and slopes Theorem (Bracci, Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D , spectral value λ > 0 and model ( S π/λ , h, z �→ z + it ) .If ( z n ) ⊂ D is a sequence converging to τ with β = n → + ∞ Arg(1 − τz n ) ∈ ( − π/ 2 , π/ 2) , lim then n →∞ Re h ( z n ) = β λ + π lim 2 λ. In particular, S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Hyperbolic semigroups: models and slopes Theorem (Bracci, Contreras, DM) Let ( ϕ t ) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂ D , spectral value λ > 0 and model ( S π/λ , h, z �→ z + it ) .If ( z n ) ⊂ D is a sequence converging to τ with β = n → + ∞ Arg(1 − τz n ) ∈ ( − π/ 2 , π/ 2) , lim then n →∞ Re h ( z n ) = β λ + π lim 2 λ. In particular, for all z ∈ D , t → + ∞ Re h ( ϕ t ( z )) = θ ( ϕ t , z ) + π lim 2 λ. λ S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

Parabolic semigroups: hyperbolic step S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) � 2 log ; 1 − α ( z 1 , z 2 ) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � Let ( ϕ t ) be a parabolic semigroup. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � Let ( ϕ t ) be a parabolic semigroup. Then, the following limit exists for every z ∈ D S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � Let ( ϕ t ) be a parabolic semigroup. Then, the following limit exists for every z ∈ D t → + ∞ k D ( ϕ t ( z ) , ϕ t +1 ( z )) . lim S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � Let ( ϕ t ) be a parabolic semigroup. Then, the following limit exists for every z ∈ D t → + ∞ k D ( ϕ t ( z ) , ϕ t +1 ( z )) . lim Moreover, the limit is always positive or it is always zero. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � Let ( ϕ t ) be a parabolic semigroup. Then, the following limit exists for every z ∈ D t → + ∞ k D ( ϕ t ( z ) , ϕ t +1 ( z )) . lim Moreover, the limit is always positive or it is always zero. If it is always positive, the semigroup is called of positive hyperbolic step. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Parabolic semigroups: hyperbolic step k D will denote the hyperbolic metric in D . That is ( z 1 , z 2 ∈ D ) k D ( z 1 , z 2 ) := 1 � 1 + α ( z 1 , z 2 ) z 1 − z 2 � � � � � 2 log ; α ( z 1 , z 2 ) = � . � � 1 − α ( z 1 , z 2 ) 1 − z 1 z 2 � Let ( ϕ t ) be a parabolic semigroup. Then, the following limit exists for every z ∈ D t → + ∞ k D ( ϕ t ( z ) , ϕ t +1 ( z )) . lim Moreover, the limit is always positive or it is always zero. If it is always positive, the semigroup is called of positive hyperbolic step. If it is always zero, the semigroup is called of zero hyperbolic step. S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

Slopes: parabolic-positive semigroups (I) S. D´ ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26