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

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Slope problems in the theory of semigroups of holomorphic self-maps of the unit disc Santiago D az-Madrigal Universidad de Sevilla Dynamical Systems: from geometry to mechanics Department of Mathematics, University of Rome Tor Vegata Rome

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Slope problems in the theory of semigroups of holomorphic self-maps of the unit disc

Santiago D´ ıaz-Madrigal Universidad de Sevilla Dynamical Systems: from geometry to mechanics Department of Mathematics, University of Rome Tor Vegata Rome (Italy), 5-8 February 2019

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Semigroups

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1}

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

1 ϕs+t = ϕt ◦ ϕs, for every t, s ≥ 0.

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

1 ϕs+t = ϕt ◦ ϕs, for every t, s ≥ 0. 2 For every z ∈ D, limt→0 ϕt(z) = z.

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

1 ϕs+t = ϕt ◦ ϕs, for every t, s ≥ 0. 2 For every z ∈ D, limt→0 ϕt(z) = z.

Fixed t ∈ [0, +∞),

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

1 ϕs+t = ϕt ◦ ϕs, for every t, s ≥ 0. 2 For every z ∈ D, limt→0 ϕt(z) = z.

Fixed t ∈ [0, +∞), the map z ∈ D → ϕt(z) is called the t-iterate of the semigroup.

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

1 ϕs+t = ϕt ◦ ϕs, for every t, s ≥ 0. 2 For every z ∈ D, limt→0 ϕt(z) = z.

Fixed t ∈ [0, +∞), the map z ∈ D → ϕt(z) is called the t-iterate of the semigroup. Fixed z ∈ D,

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Semigroups

A semigroup in D := {z ∈ C : |z| < 1} is any family (ϕt) of holomorphic self-maps of D, verifying the following two conditions:

1 ϕs+t = ϕt ◦ ϕs, for every t, s ≥ 0. 2 For every z ∈ D, limt→0 ϕt(z) = z.

Fixed t ∈ [0, +∞), the map z ∈ D → ϕt(z) is called the t-iterate of the semigroup. Fixed z ∈ D, the map t ∈ [0, +∞) → ϕt(z) is called the z-trajectory of the semigroup.

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Vector fields

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Vector fields

There exists a unique holomorphic function G : D → C,

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt)

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z is exactly t ∈ [0, +∞) → ϕt(z)

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z is exactly t ∈ [0, +∞) → ϕt(z) that is, the z-trajectory.

  • S. D´

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z is exactly t ∈ [0, +∞) → ϕt(z) that is, the z-trajectory. Dynamical questions:

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z is exactly t ∈ [0, +∞) → ϕt(z) that is, the z-trajectory. Dynamical questions:

1 Asymptotic behaviour of the trajectories.

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z is exactly t ∈ [0, +∞) → ϕt(z) that is, the z-trajectory. Dynamical questions:

1 Asymptotic behaviour of the trajectories. 2 Slope analysis of those trajectories.

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Vector fields

There exists a unique holomorphic function G : D → C, which is called the vector field of the semigroup (ϕt) such that, for every z ∈ D, the solution of the Cauchy problem

  • w′ = G(w)

w(0) = z is exactly t ∈ [0, +∞) → ϕt(z) that is, the z-trajectory. Dynamical questions:

1 Asymptotic behaviour of the trajectories. 2 Slope analysis of those trajectories. 3 Others: poles and fractional singularities of the vector field, rate of

convergence, contact arcs, synchronization formulae for fixed points, ....

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Asymptotic behaviour

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity;

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(z)).

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(z)). That is, trajectories are rotations inside D.

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(z)). That is, trajectories are rotations inside D.

3 There exists τ ∈ D

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(z)). That is, trajectories are rotations inside D.

3 There exists τ ∈ D (unique and called the Denjoy-Wolff point of

the semigroup)

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(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, limt→+∞ ϕt(z) = τ.

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(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, limt→+∞ ϕt(z) = τ. Semigroups verifying (1), (2) or (3) with τ ∈ D are called elliptic.

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Asymptotic behaviour

Theorem (The continuous Denjoy-Wolff theorem)

Three possible situations can happen:

1 Every iterate ϕt is the identity; that is, every trajectory is

stationary.

2 There exist ω ∈ R \ {0} and an automorphism T of D such that

ϕt(z) = T −1(eitωT(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, limt→+∞ ϕt(z) = τ. Semigroups verifying (1), (2) or (3) with τ ∈ D are called elliptic. Semigroups verifying (3) with τ ∈ ∂D are called non-elliptic.

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Hyperbolic and parabolic semigroups

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Hyperbolic and parabolic semigroups

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D.

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Hyperbolic and parabolic semigroups

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. Then, there exists λ ∈ [0, +∞)

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Hyperbolic and parabolic semigroups

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. Then, there exists λ ∈ [0, +∞), called the spectral value of the semigroup (ϕt)

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Hyperbolic and parabolic semigroups

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. Then, there exists λ ∈ [0, +∞), called the spectral value of the semigroup (ϕt) such that lim

r→1 ϕ′ t(rτ) = e−λt,

t ≥ 0.

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Hyperbolic and parabolic semigroups

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. Then, there exists λ ∈ [0, +∞), called the spectral value of the semigroup (ϕt) such that lim

r→1 ϕ′ t(rτ) = e−λt,

t ≥ 0. Hyperbolic semigroups are those with λ > 0.

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Hyperbolic and parabolic semigroups

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. Then, there exists λ ∈ [0, +∞), called the spectral value of the semigroup (ϕt) such that lim

r→1 ϕ′ t(rτ) = e−λt,

t ≥ 0. Hyperbolic semigroups are those with λ > 0. Parabolic semigroups are those with λ = 0.

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Slopes in the context of non-elliptic semigroups (I)

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D.

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. The set of slopes of (ϕt) when arriving at τ

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. The set of slopes of (ϕt) when arriving at τ, and denoted as Slope[ϕt(z), τ]

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. The set of slopes of (ϕt) when arriving at τ, and denoted as Slope[ϕt(z), τ] ⊂

  • −π

2 , π 2

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. The set of slopes of (ϕt) when arriving at τ, and denoted as Slope[ϕt(z), τ] ⊂

  • −π

2 , π 2

  • is defined as the cluster set at t = +∞ of the curve

t ∈ [0, +∞) → Arg(1 − τϕt(z)).

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. The set of slopes of (ϕt) when arriving at τ, and denoted as Slope[ϕt(z), τ] ⊂

  • −π

2 , π 2

  • is defined as the cluster set at t = +∞ of the curve

t ∈ [0, +∞) → Arg(1 − τϕt(z)). In other words, θ ∈

−π

2 , π 2

belongs to Slope[ϕt(z), τ]

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Slopes in the context of non-elliptic semigroups (I)

Let (ϕt) be a non-elliptic semigroup with Denjoy-Wolff point τ ∈ ∂D. The set of slopes of (ϕt) when arriving at τ, and denoted as Slope[ϕt(z), τ] ⊂

  • −π

2 , π 2

  • is defined as the cluster set at t = +∞ of the curve

t ∈ [0, +∞) → Arg(1 − τϕt(z)). In other words, θ ∈

−π

2 , π 2

belongs to Slope[ϕt(z), τ] if and only if

there exists a sequence (tn) ⊂ [0, +∞) converging to +∞ such that lim

n→∞ Arg(1 − τϕtn(z)) = θ.

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Slopes in the context of non-elliptic semigroups (II)

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Slopes in the context of non-elliptic semigroups (II)

Slope[ϕt(z), τ] is either a point or a closed subinterval of

−π

2 , π 2

.

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Slopes in the context of non-elliptic semigroups (II)

Slope[ϕt(z), τ] is either a point or a closed subinterval of

−π

2 , π 2

.

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Slopes in the context of non-elliptic semigroups (III)

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Slopes in the context of non-elliptic semigroups (III)

Concerning oscillations,

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Slopes in the context of non-elliptic semigroups (III)

Concerning oscillations, it is important to consider when they can be considered “moderated”

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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).

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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´

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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[ϕt(z), τ] is a closed subinterval of (−π/2, π/2).

  • S. D´

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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[ϕt(z), τ] is a closed subinterval of (−π/2, π/2). (ϕt(z)) converges non-tangentially to τ

  • S. D´

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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[ϕt(z), τ] is a closed subinterval of (−π/2, π/2). (ϕt(z)) converges non-tangentially to τ if Slope[ϕt(z), τ] is a singleton or a closed subinterval of (−π/2, π/2).

  • S. D´

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Slopes: hyperbolic semigroups

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D.

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D,

  • S. D´

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that

  • S. D´

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that Slope[ϕt(z), τ] = {θ(z)}.

  • S. D´

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that Slope[ϕt(z), τ] = {θ(z)}. Moreover

  • S. D´

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that Slope[ϕt(z), τ] = {θ(z)}. Moreover

1 θ : D → R is an harmonic function.

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that Slope[ϕt(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´

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Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that Slope[ϕt(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(z1) = θ(z2) if and only if there is t ≥ 0 such that either

ϕt(z1) = z2 or ϕt(z2) = z1.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

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SLIDE 69

Slopes: hyperbolic semigroups

Theorem (Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then. for every z ∈ D, there exists θ(ϕt, z) = θ(z) ∈ (−π/2, π/2) such that Slope[ϕt(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(z1) = θ(z2) if and only if there is t ≥ 0 such that either

ϕt(z1) = z2 or ϕt(z2) = z1.

4 θ (essentially) determines the hyperbolic semigroup (ϕt).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 9 / 26

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SLIDE 70

Linear models for hyperbolic semigroups (I)

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ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 71

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 72

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 73

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 74

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

1 Sπ/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

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SLIDE 75

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

1 Sπ/λ is the open strip {x + iy ∈ C : x ∈ (0, π/λ), y ∈ R}. 2 h is a univalent function from D into Sπ/λ

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 76

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

1 Sπ/λ is the open strip {x + iy ∈ C : x ∈ (0, π/λ), y ∈ R}. 2 h is a univalent function from D into Sπ/λ

(hence h(D) is a simply connected domain).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 77

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

1 Sπ/λ is the open strip {x + iy ∈ C : x ∈ (0, π/λ), y ∈ R}. 2 h is a univalent function from D into Sπ/λ

(hence h(D) is a simply connected domain).

3 h ◦ ϕt(z) = h(z) + it,

z ∈ D, t ≥ 0

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 10 / 26

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SLIDE 78

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

1 Sπ/λ is the open strip {x + iy ∈ C : x ∈ (0, π/λ), y ∈ R}. 2 h is a univalent function from D into Sπ/λ

(hence h(D) is a simply connected domain).

3 h ◦ ϕt(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

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SLIDE 79

Linear models for hyperbolic semigroups (I)

Every hyperbolic semigroup (ϕt) with spectral value λ > 0has an essentially unique model (Sπ/λ, h, z → z + it). This means:

1 Sπ/λ is the open strip {x + iy ∈ C : x ∈ (0, π/λ), y ∈ R}. 2 h is a univalent function from D into Sπ/λ

(hence h(D) is a simply connected domain).

3 h ◦ ϕt(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

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SLIDE 80

Linear models for hyperbolic semigroups (II)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 11 / 26

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SLIDE 81

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

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SLIDE 82

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

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SLIDE 83

Hyperbolic semigroups: models and slopes

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ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 84

Hyperbolic semigroups: models and slopes

Theorem (Bracci, Contreras, DM)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 85

Hyperbolic semigroups: models and slopes

Theorem (Bracci, Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0 and model (Sπ/λ, h, z → z + it).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 86

Hyperbolic semigroups: models and slopes

Theorem (Bracci, Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0 and model (Sπ/λ, h, z → z + it).If (zn) ⊂ D is a sequence converging to τ with β = lim

n→+∞ Arg(1 − τzn) ∈ (−π/2, π/2),

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 87

Hyperbolic semigroups: models and slopes

Theorem (Bracci, Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0 and model (Sπ/λ, h, z → z + it).If (zn) ⊂ D is a sequence converging to τ with β = lim

n→+∞ Arg(1 − τzn) ∈ (−π/2, π/2),

then lim

n→∞ Re h(zn) = β

λ + π 2λ.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 88

Hyperbolic semigroups: models and slopes

Theorem (Bracci, Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0 and model (Sπ/λ, h, z → z + it).If (zn) ⊂ D is a sequence converging to τ with β = lim

n→+∞ Arg(1 − τzn) ∈ (−π/2, π/2),

then lim

n→∞ Re h(zn) = β

λ + π 2λ. In particular,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 89

Hyperbolic semigroups: models and slopes

Theorem (Bracci, Contreras, DM)

Let (ϕt) be a hyperbolic semigroup in D with Denjoy-Wolff point τ ∈ ∂D, spectral value λ > 0 and model (Sπ/λ, h, z → z + it).If (zn) ⊂ D is a sequence converging to τ with β = lim

n→+∞ Arg(1 − τzn) ∈ (−π/2, π/2),

then lim

n→∞ Re h(zn) = β

λ + π 2λ. In particular, for all z ∈ D, lim

t→+∞ Re h(ϕt(z)) = θ(ϕt, z)

λ + π 2λ.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 12 / 26

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SLIDE 90

Parabolic semigroups: hyperbolic step

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 91

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 92

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ;
  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 93

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .
  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 94

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .

Let (ϕt) be a parabolic semigroup.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 95

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .

Let (ϕt) 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

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SLIDE 96

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .

Let (ϕt) be a parabolic semigroup. Then, the following limit exists for every z ∈ D lim

t→+∞ kD(ϕt(z), ϕt+1(z)).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 97

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .

Let (ϕt) be a parabolic semigroup. Then, the following limit exists for every z ∈ D lim

t→+∞ kD(ϕt(z), ϕt+1(z)).

Moreover, the limit is always positive or it is always zero.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 13 / 26

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SLIDE 98

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .

Let (ϕt) be a parabolic semigroup. Then, the following limit exists for every z ∈ D lim

t→+∞ kD(ϕt(z), ϕt+1(z)).

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

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SLIDE 99

Parabolic semigroups: hyperbolic step

kD will denote the hyperbolic metric in D. That is (z1, z2 ∈ D) kD(z1, z2) := 1 2 log

1 + α(z1, z2)

1 − α(z1, z2)

  • ; α(z1, z2) =
  • z1 − z2

1 − z1z2

  • .

Let (ϕt) be a parabolic semigroup. Then, the following limit exists for every z ∈ D lim

t→+∞ kD(ϕt(z), ϕt+1(z)).

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

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SLIDE 100

Slopes: parabolic-positive semigroups (I)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26

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SLIDE 101

Slopes: parabolic-positive semigroups (I)

Theorem (Contreras, DM)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26

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SLIDE 102

Slopes: parabolic-positive semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-positive semigroup.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26

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SLIDE 103

Slopes: parabolic-positive semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-positive semigroup. Then, either for every z ∈ D Slope[ϕt(z), τ] =

π

2

  • .
  • r,
  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26

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SLIDE 104

Slopes: parabolic-positive semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-positive semigroup. Then, either for every z ∈ D Slope[ϕt(z), τ] =

π

2

  • .
  • r, for every z ∈ D

Slope[ϕt(z), τ] =

  • −π

2

  • .
  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26

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SLIDE 105

Slopes: parabolic-positive semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-positive semigroup. Then, either for every z ∈ D Slope[ϕt(z), τ] =

π

2

  • .
  • r, for every z ∈ D

Slope[ϕt(z), τ] =

  • −π

2

  • .

In other words, the trajectories “are asymptotically tangential” to the boundary of D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 14 / 26

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SLIDE 106

Slopes: parabolic-positive semigroups (II)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 15 / 26

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SLIDE 107

Slopes: parabolic-positive semigroups (II)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 15 / 26

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SLIDE 108

Slopes: parabolic-zero semigroups (I)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 109

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 110

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 111

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z1, z2 ∈ D, Slope[ϕt(z1), τ] = Slope[ϕt(z2), τ].

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 112

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z1, z2 ∈ D, Slope[ϕt(z1), τ] = Slope[ϕt(z2), τ]. The analysis of the slopes does not depend on the point z.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 113

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z1, z2 ∈ D, Slope[ϕt(z1), τ] = Slope[ϕt(z2), τ]. The analysis of the slopes does not depend on the point z.

Theorem (Betsakos,Contreras,DM,Gumenyuk,Kelgiannis)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 114

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z1, z2 ∈ D, Slope[ϕt(z1), τ] = Slope[ϕt(z2), τ]. The analysis of the slopes does not depend on the point z.

Theorem (Betsakos,Contreras,DM,Gumenyuk,Kelgiannis)

Given −π/2 ≤ θ1 ≤ θ2 ≤ π/2,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 115

Slopes: parabolic-zero semigroups (I)

Theorem (Contreras, DM)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then, for every z1, z2 ∈ D, Slope[ϕt(z1), τ] = Slope[ϕt(z2), τ]. The analysis of the slopes does not depend on the point z.

Theorem (Betsakos,Contreras,DM,Gumenyuk,Kelgiannis)

Given −π/2 ≤ θ1 ≤ θ2 ≤ π/2, there exists a parabolic-zero semigroup (ϕt) such that Slope[ϕt(z), τ] = [θ1, θ2].

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 16 / 26

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SLIDE 116

Slopes: parabolic-zero semigroups (II)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

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SLIDE 117

Slopes: parabolic-zero semigroups (II)

Because of the example,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

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SLIDE 118

Slopes: parabolic-zero semigroups (II)

Because of the example, two questions (at least) appear naturally:

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

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SLIDE 119

Slopes: parabolic-zero semigroups (II)

Because of the example, two questions (at least) appear naturally:

1 When does Slope[ϕt(z), τ] reduce to a point?

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

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SLIDE 120

Slopes: parabolic-zero semigroups (II)

Because of the example, two questions (at least) appear naturally:

1 When does Slope[ϕt(z), τ] reduce to a point?

(that is, when there are well defined slopes).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

slide-121
SLIDE 121

Slopes: parabolic-zero semigroups (II)

Because of the example, two questions (at least) appear naturally:

1 When does Slope[ϕt(z), τ] reduce to a point?

(that is, when there are well defined slopes).

2 When does Slope[ϕt(z), τ] is a closed subinterval of (−π/2, π/2) or a

singleton?

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

slide-122
SLIDE 122

Slopes: parabolic-zero semigroups (II)

Because of the example, two questions (at least) appear naturally:

1 When does Slope[ϕt(z), τ] reduce to a point?

(that is, when there are well defined slopes).

2 When does Slope[ϕt(z), τ] is a closed subinterval of (−π/2, π/2) or a

singleton? (that is, when there is non-tangential convergence).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

slide-123
SLIDE 123

Slopes: parabolic-zero semigroups (II)

Because of the example, two questions (at least) appear naturally:

1 When does Slope[ϕt(z), τ] reduce to a point?

(that is, when there are well defined slopes).

2 When does Slope[ϕt(z), τ] is a closed subinterval of (−π/2, π/2) or a

singleton? (that is, when there is non-tangential convergence).

In the rest of the talk, we mainly treat the second question.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 17 / 26

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SLIDE 124

Linear models for parabolic-zero semigroups (I)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-125
SLIDE 125

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-126
SLIDE 126

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

1 h is a univalent function from D into C

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-127
SLIDE 127

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

1 h is a univalent function from D into C

(h(D) is a simply connected domain different from the whole C).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-128
SLIDE 128

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

1 h is a univalent function from D into C

(h(D) is a simply connected domain different from the whole C).

2 h ◦ ϕt(z) = h(z) + it,

z ∈ D, t ≥ 0.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-129
SLIDE 129

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

1 h is a univalent function from D into C

(h(D) is a simply connected domain different from the whole C).

2 h ◦ ϕt(z) = h(z) + it,

z ∈ D, t ≥ 0. (again h(D) + it ⊂ h(D)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-130
SLIDE 130

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

1 h is a univalent function from D into C

(h(D) is a simply connected domain different from the whole C).

2 h ◦ ϕt(z) = h(z) + it,

z ∈ D, t ≥ 0. (again h(D) + it ⊂ h(D) − → h(D) is a starlike at infinite domain).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

slide-131
SLIDE 131

Linear models for parabolic-zero semigroups (I)

Every parabolic semigroup (ϕt) of zero hyperbolic step has a (essentially) unique model (C, h, z → z + it) where

1 h is a univalent function from D into C

(h(D) is a simply connected domain different from the whole C).

2 h ◦ ϕt(z) = h(z) + it,

z ∈ D, t ≥ 0. (again h(D) + it ⊂ h(D) − → h(D) is a starlike at infinite domain).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 18 / 26

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SLIDE 132

Linear models for parabolic-zero semigroups (II)

  • S. D´

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SLIDE 133

Linear models for parabolic-zero semigroups (II)

Again, we look at geometrical aspects of Ω := h(D) to attack the problem of non-tangential convergence.

  • S. D´

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SLIDE 134

Linear models for parabolic-zero semigroups (II)

Again, we look at geometrical aspects of Ω := h(D) to attack the problem of non-tangential convergence.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 19 / 26

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SLIDE 135

Parabolic-zero semigroups: models and slopes (I)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 136

Parabolic-zero semigroups: models and slopes (I)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 137

Parabolic-zero semigroups: models and slopes (I)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then, for t ≥ 0 and p ∈ Ω, we define

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 138

Parabolic-zero semigroups: models and slopes (I)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then, for t ≥ 0 and p ∈ Ω, we define δ+

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≥ Re p}},

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 139

Parabolic-zero semigroups: models and slopes (I)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then, for t ≥ 0 and p ∈ Ω, we define δ+

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≥ Re p}},

and δ−

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≤ Re p}}.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 140

Parabolic-zero semigroups: models and slopes (I)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then, for t ≥ 0 and p ∈ Ω, we define δ+

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≥ Re p}},

and δ−

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≤ Re p}}.

For t large enough and asymptotically,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 141

Parabolic-zero semigroups: models and slopes (I)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then, for t ≥ 0 and p ∈ Ω, we define δ+

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≥ Re p}},

and δ−

Ω,p(t) := min{t, inf{|z − (p + it)| : z ∈ ∂Ω, Re z ≤ Re p}}.

For t large enough and asymptotically, δ+

Ω,p(t) and δ− Ω,p(t) measure

(in a normalized way) the symmetrical aspect of Ω with respect to the trajectory t → p + it.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 20 / 26

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SLIDE 142

Parabolic-zero semigroups: models and slopes (II)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 21 / 26

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SLIDE 143

Parabolic-zero semigroups: models and slopes (II)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 21 / 26

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SLIDE 144

Parabolic-zero semigroups: models and slopes (II)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 21 / 26

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SLIDE 145

Parabolic-zero semigroups: models and slopes (II)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then the following are equivalent:

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 21 / 26

slide-146
SLIDE 146

Parabolic-zero semigroups: models and slopes (II)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then the following are equivalent:

1 For some (resp. all) z ∈ D, (ϕt(z)) converges non-tangentially to

the point τ.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 21 / 26

slide-147
SLIDE 147

Parabolic-zero semigroups: models and slopes (II)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). Then the following are equivalent:

1 For some (resp. all) z ∈ D, (ϕt(z)) converges non-tangentially to

the point τ.

2 For some (resp. all) p ∈ Ω there exist 0 < c < C and t0 ≥ 0 such

that for all t ≥ t0 cδ+

Ω,p(t) ≤ δ− Ω,p(t) ≤ Cδ+ Ω,p(t).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 21 / 26

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SLIDE 148

Parabolic-zero semigroups: models and slopes (III)

  • S. D´

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SLIDE 149

Parabolic-zero semigroups: models and slopes (III)

  • S. D´

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SLIDE 150

Parabolic-zero semigroups: models and slopes (IV)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 23 / 26

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SLIDE 151

Parabolic-zero semigroups: models and slopes (IV)

Theorem (Bracci,Contreras,DM,Gaussier)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 23 / 26

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SLIDE 152

Parabolic-zero semigroups: models and slopes (IV)

Theorem (Bracci,Contreras,DM,Gaussier)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 23 / 26

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SLIDE 153

Parabolic-zero semigroups: models and slopes (IV)

Theorem (Bracci,Contreras,DM,Gaussier)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). If Ω contains a (small) angular sector,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 23 / 26

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SLIDE 154

Parabolic-zero semigroups: models and slopes (IV)

Theorem (Bracci,Contreras,DM,Gaussier)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). If Ω contains a (small) angular sector, then (ϕt(z)) converges non-tangentially to the point τ for all z ∈ D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 23 / 26

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SLIDE 155

Parabolic-zero semigroups: models and slopes (IV)

Theorem (Bracci,Contreras,DM,Gaussier)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it) and let Ω := h(D). If Ω contains a (small) angular sector, then (ϕt(z)) converges non-tangentially to the point τ for all z ∈ D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 23 / 26

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SLIDE 156

Parabolic-zero semigroups: models and slopes (V)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 157

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 158

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it).

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 159

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it). Let Ω := h(D). Then:

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 160

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it). Let Ω := h(D). Then:

1 Slope[ϕt(z), τ] = {π

2 } for some (resp. all) z ∈ D

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 161

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it). Let Ω := h(D). Then:

1 Slope[ϕt(z), τ] = {π

2 } for some (resp. all) z ∈ D if and only if for

some (resp. all) p ∈ Ω, lim

t→+∞

δ+

Ω,p(t)

δ−

Ω,p(t) = 0.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 162

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it). Let Ω := h(D). Then:

1 Slope[ϕt(z), τ] = {π

2 } for some (resp. all) z ∈ D if and only if for

some (resp. all) p ∈ Ω, lim

t→+∞

δ+

Ω,p(t)

δ−

Ω,p(t) = 0.

2 Slope[ϕt(z), τ] = {−π

2 } for some (resp. all) z ∈ D

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 24 / 26

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SLIDE 163

Parabolic-zero semigroups: models and slopes (V)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D and model (C, h, z → z + it). Let Ω := h(D). Then:

1 Slope[ϕt(z), τ] = {π

2 } for some (resp. all) z ∈ D if and only if for

some (resp. all) p ∈ Ω, lim

t→+∞

δ+

Ω,p(t)

δ−

Ω,p(t) = 0.

2 Slope[ϕt(z), τ] = {−π

2 } for some (resp. all) z ∈ D if and only if for

some (resp. all) p ∈ Ω, lim

t→+∞

δ+

Ω,p(t)

δ−

Ω,p(t) = +∞.

  • S. D´

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SLIDE 164

Some notions from hyperbolic geometry

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SLIDE 165

Some notions from hyperbolic geometry

Geodesics in D:

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SLIDE 166

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D

  • S. D´

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SLIDE 167

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).
  • S. D´

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SLIDE 168

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 25 / 26

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SLIDE 169

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D if for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) = kD(γ(s), γ(t)),

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 25 / 26

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SLIDE 170

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D if for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) = kD(γ(s), γ(t)), where ℓD(γ; [s, t]) :=

t

s |γ′(u)| 1−|γ(u)|2 du is the hyperbolic length of γ

restricted to [s, t].

  • S. D´

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SLIDE 171

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D if for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) = kD(γ(s), γ(t)), where ℓD(γ; [s, t]) :=

t

s |γ′(u)| 1−|γ(u)|2 du is the hyperbolic length of γ

restricted to [s, t]. In other words,

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 25 / 26

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SLIDE 172

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D if for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) = kD(γ(s), γ(t)), where ℓD(γ; [s, t]) :=

t

s |γ′(u)| 1−|γ(u)|2 du is the hyperbolic length of γ

restricted to [s, t]. In other words, geodesics are the “best” way in the hyperbolic sense to move from one point to another.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 25 / 26

slide-173
SLIDE 173

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D if for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) = kD(γ(s), γ(t)), where ℓD(γ; [s, t]) :=

t

s |γ′(u)| 1−|γ(u)|2 du is the hyperbolic length of γ

restricted to [s, t]. In other words, geodesics are the “best” way in the hyperbolic sense to move from one point to another. A Lipschitz continuous curve γ : [a, b] → D is a quasi-geodesic in D

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 25 / 26

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SLIDE 174

Some notions from hyperbolic geometry

Geodesics in D: Lipschitz continuous curves supported on hyperbolic lines in D (diameters and circle arcs which cut

  • rthogonally ∂D).

A Lipschitz continuous curve γ : [a, b] → D is a geodesic in D if for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) = kD(γ(s), γ(t)), where ℓD(γ; [s, t]) :=

t

s |γ′(u)| 1−|γ(u)|2 du is the hyperbolic length of γ

restricted to [s, t]. In other words, geodesics are the “best” way in the hyperbolic sense to move from one point to another. A Lipschitz continuous curve γ : [a, b] → D is a quasi-geodesic in D if for some A ≥ 1 and B ≥ 0 and for every a ≤ s < t ≤ b, ℓD(γ; [s, t]) ≤ AkD(γ(s), γ(t)) + B.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 25 / 26

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SLIDE 175

Parabolic-zero semigroups: models and slopes (VI)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26

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SLIDE 176

Parabolic-zero semigroups: models and slopes (VI)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26

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SLIDE 177

Parabolic-zero semigroups: models and slopes (VI)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26

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SLIDE 178

Parabolic-zero semigroups: models and slopes (VI)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then the following are equivalent:

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26

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SLIDE 179

Parabolic-zero semigroups: models and slopes (VI)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then the following are equivalent:

1 For some (resp. all) z ∈ D, (ϕt(z)) converges non-tangentially to

the point τ.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26

slide-180
SLIDE 180

Parabolic-zero semigroups: models and slopes (VI)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then the following are equivalent:

1 For some (resp. all) z ∈ D, (ϕt(z)) converges non-tangentially to

the point τ.

2 For some (resp. all) z ∈ D, the curve

t ∈ [0, +∞) → ϕt(z) is a quasi-geodesic in D.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26

slide-181
SLIDE 181

Parabolic-zero semigroups: models and slopes (VI)

Theorem (Bracci,Contreras,DM,Gaussier,Zimmer)

Let (ϕt) be a parabolic-zero semigroup in D with Denjoy-Wolff point τ ∈ ∂D. Then the following are equivalent:

1 For some (resp. all) z ∈ D, (ϕt(z)) converges non-tangentially to

the point τ.

2 For some (resp. all) z ∈ D, the curve

t ∈ [0, +∞) → ϕt(z) is a quasi-geodesic in D. A similar result is true for general non-elliptic semigroups.

  • S. D´

ıaz-Madrigal Universidad de Sevilla February 13, 2019 26 / 26