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Topology and dynamics on the boundary of two-dimensional domains - - PowerPoint PPT Presentation

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Topology and dynamics on the boundary of two-dimensional domains Meysam Nassiri IPM - Institute for Research in Fundamental Sciences Tehran Joint work with Andres Koropecki and Patrice Le Calvez Meysam Nassiri (IPM) Boundary dynamics and

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Topology and dynamics on the boundary of two-dimensional domains

Meysam Nassiri

IPM - Institute for Research in Fundamental Sciences Tehran Joint work with Andres Koropecki and Patrice Le Calvez

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Basic Problem

f : S → S homeomorphism of an orientable surface; U ⊂ S invariant domain; Describe the dynamics in the boundary of U.

◮ Existence of periodic points in ∂ U ◮ Topological restrictions imposed by the dynamics of f |∂ U. Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Simplest setting

f : R2 → R2 orientation-preserving homeomorphism; U ⊂ R2 bounded, f -invariant, open, simply connected.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Simplest setting

f : R2 → R2 orientation-preserving homeomorphism; U ⊂ R2 bounded, f -invariant, open, simply connected.

Question

Existence of periodic point of f in ∂ U ? Any necessary and sufficient condition?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Simplest setting

f : R2 → R2 orientation-preserving homeomorphism; U ⊂ R2 bounded, f -invariant, open, simply connected.

Question

Existence of periodic point of f in ∂ U ? Any necessary and sufficient condition?

Simplest simplest case:

∂ U is a circle (so U ≃ D)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Simplest setting

f : R2 → R2 orientation-preserving homeomorphism; U ⊂ R2 bounded, f -invariant, open, simply connected.

Question

Existence of periodic point of f in ∂ U ? Any necessary and sufficient condition?

Simplest simplest case:

∂ U is a circle (so U ≃ D) = ⇒ f |∂ U is a circle homeomorphism

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Simplest setting

f : R2 → R2 orientation-preserving homeomorphism; U ⊂ R2 bounded, f -invariant, open, simply connected.

Question

Existence of periodic point of f in ∂ U ? Any necessary and sufficient condition?

Simplest simplest case:

∂ U is a circle (so U ≃ D) = ⇒ f |∂ U is a circle homeomorphism = ⇒ Poincar´ e Theory. Key: Rotation number!

Theorem (Poincar´ e)

∃ periodic point ⇐ ⇒ rotation number of f |∂ U is rational.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Problem

Usually ∂ U is not circle!

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Problem

Usually ∂ U is not circle! Not even similar. ∂ U can have very very complicated topology!

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Problem

Usually ∂ U is not circle! Not even similar. ∂ U can have very very complicated topology!

  • may have points inaccessible from U,
  • can be nowhere locally connected,
  • worse things (e.g. an hereditarily indecomposable continuum)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Problem

Usually ∂ U is not circle! Not even similar. ∂ U can have very very complicated topology!

  • may have points inaccessible from U,
  • can be nowhere locally connected,
  • worse things (e.g. an hereditarily indecomposable continuum)
  • these are not isolated or infrequent, independently of regularity.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

Poincar´ e-like theory for ∂ U ? How to associate a rotation number to f and U ?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

Poincar´ e-like theory for ∂ U ? How to associate a rotation number to f and U ? How to associate a circle homeomorphism to f and U ?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

Poincar´ e-like theory for ∂ U ? How to associate a rotation number to f and U ? How to associate a circle homeomorphism to f and U ?

Idea

Compactify U by adding an “ideal” circle (in a sensible way)

  • U := U ⊔ S1

with a suitable topology such that U ≃ D. Hopefully, f |U extends to f : U → U. Define the rotation number ρ(f , U) := ρ( f |S1).

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

Poincar´ e-like theory for ∂ U ? How to associate a rotation number to f and U ? How to associate a circle homeomorphism to f and U ?

Idea

Compactify U by adding an “ideal” circle (in a sensible way)

  • U := U ⊔ S1

with a suitable topology such that U ≃ D. Hopefully, f |U extends to f : U → U. Define the rotation number ρ(f , U) := ρ( f |S1).

Cartwright-Littlewood, 1951

  • U = Carath´

eodory’s prime ends compactification ρ(f , U) = Prime ends rotation number.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

How is the relation between two dynamics: f has a periodic point in ∂ U

?

⇐ = = ⇒ ρ(f , U) ∈ Q

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

How is the relation between two dynamics: f has a periodic point in ∂ U

?

⇐ = = ⇒ ρ(f , U) ∈ Q Answer: No in both directions!

Figure : ρ = 0 and Fix(f |∂ U) = ∅ Figure : ρ / ∈ Q and Fix(f |∂ U) = circle

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

How is the relation between two dynamics: f has a periodic point in ∂ U

?

⇐ = = ⇒ ρ(f , U) ∈ Q Answer: No in both directions!

Figure : ρ = 0 and Fix(f |∂ U) = ∅ Figure : ρ / ∈ Q and Fix(f |∂ U) = circle

  • Note: Both examples have attracting regions near the boundary.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Question

How is the relation between two dynamics: f has a periodic point in ∂ U

?

⇐ = = ⇒ ρ(f , U) ∈ Q Answer: No in both directions!

Figure : ρ = 0 and Fix(f |∂ U) = ∅ Figure : ρ / ∈ Q and Fix(f |∂ U) = circle

  • Note: Both examples have attracting regions near the boundary.
  • Not possible if f preserves area (or nonwandering)....

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Consequences of the rotation number

f : R2 → R2 homeomorphism U ⊂ R2 bounded, simply connected, open, f -invariant f is nonwandering (e.g. area-preserving) in U.

Theorem (Cartwright-Littlewood, 1951)

ρ(f , U) ∈ Q = ⇒ ∃ periodic point in ∂ U

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Consequences of the rotation number

f : R2 → R2 homeomorphism U ⊂ R2 bounded, simply connected, open, f -invariant f is nonwandering (e.g. area-preserving) in U.

Theorem (Cartwright-Littlewood, 1951)

ρ(f , U) ∈ Q = ⇒ ∃ periodic point in ∂ U Refinements of this result: Barge-Gillette 1991, Barge-Kuperberg 1998, Ortega-Ruiz del Portal 2011

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction

Consequences of the rotation number

f : R2 → R2 homeomorphism U ⊂ R2 bounded, simply connected, open, f -invariant f is nonwandering (e.g. area-preserving) in U.

Theorem (Cartwright-Littlewood, 1951)

ρ(f , U) ∈ Q = ⇒ ∃ periodic point in ∂ U Refinements of this result: Barge-Gillette 1991, Barge-Kuperberg 1998, Ortega-Ruiz del Portal 2011

Opposite direction? What if ρ / ∈ Q?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Results

f : R2 → R2 homeomorphism U ⊂ R2 bounded, simply connected, open, f -invariant f is nonwandering (e.g. area-preserving).

Theorem A (Converse of [C-L])

ρ(f , U) / ∈ Q = ⇒ ∄ periodic point in ∂ U

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Results

f : R2 → R2 homeomorphism U ⊂ R2 bounded, simply connected, open, f -invariant f is nonwandering (e.g. area-preserving).

Theorem A (Converse of [C-L])

ρ(f , U) / ∈ Q = ⇒ ∄ periodic point in ∂ U and ∂ U is annular.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Results

f : R2 → R2 homeomorphism U ⊂ R2 simply connected, open, f -invariant f is nonwandering.

Theorem A’

ρ(f , U) = 0 = ⇒ ∄ fixed point in ∂ U.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Results

f : R2 → R2 homeomorphism U ⊂ R2 simply connected, open, f -invariant f is nonwandering.

Theorem A’

ρ(f , U) = 0 = ⇒ ∄ fixed point in ∂ U. Moreover: if U is unbounded, ρ(f , U) = 0 = ⇒ ∄ fixed point in R2 \ U.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Question

Still true for an arbitrary surface S?

U

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Question

Still true for an arbitrary surface S?

U

Figure : a simply connected open set

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Homeomorphisms of plane

Question

Still true for an arbitrary surface S? U

Figure : unique fixed point in ∂ U, surface = S2

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Theorem B (on closed surfaces)

f nonwandering homeomorphism of a closed orientable surface S, U ⊂ S open, f -invariant, simply connected. ρ(f , U) / ∈ Q One of these two holds:

1 ∂ U contains a unique fixed point and no other periodic points

S = Sphere, U is dense in S, ∂ U = S \ U cellular continuum, or

2 ∂ U is aperiodic contractible annular continuum.

U

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Figure : Impossible! if ρ(f , U) / ∈ Q.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Theorem B’

Theorem B extends to surfaces of finite type (non-compact); any invariant connected open set U; a ∂-nonwandering condition (instead of nonwandering)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Theorem B’

Theorem B extends to surfaces of finite type (non-compact); any invariant connected open set U; a ∂-nonwandering condition (instead of nonwandering)

Remark

The ∂-nonwandering condition holds if f is a holomorphic diffeomorphism.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Theorem B’

Theorem B extends to surfaces of finite type (non-compact); any invariant connected open set U; a ∂-nonwandering condition (instead of nonwandering)

Remark

The ∂-nonwandering condition holds if f is a holomorphic diffeomorphism. = ⇒ consequences in one-dimensional holomorphic dynamics.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Application for generic area-preserving diffeos

Theorem [Mather ’81]

f a C r-generic area preserving diffeomorphism (r ≥ 16), U periodic complementary domain, = ⇒ prime ends rotation numbers of U are irrational at each end. Example: p ∈ Perh(f ), U = connected component of S \ W s(p).

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Surface homeomorphisms

Application for generic area-preserving diffeos

Theorem [Mather ’81]

f a C r-generic area preserving diffeomorphism (r ≥ 16), U periodic complementary domain, = ⇒ prime ends rotation numbers of U are irrational at each end. Example: p ∈ Perh(f ), U = connected component of S \ W s(p). Dynamical consequences?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Theorem C

f a C r-generic area preserving diffeomorphism (r ≥ 1), U periodic complementary domain,

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Theorem C

f a C r-generic area preserving diffeomorphism (r ≥ 1), U periodic complementary domain, Then,

1 no periodic points in ∂ U, 2 ∂ U = finite disjoint union of aperiodic annular continua. Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Theorem C

f a C r-generic area preserving diffeomorphism (r ≥ 1), U periodic complementary domain, Then,

1 no periodic points in ∂ U, 2 ∂ U = finite disjoint union of aperiodic annular continua.

Example: p ∈ Perh(f ), U = connected component of S \ W s(p).

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Theorem C

f a C r-generic area preserving diffeomorphism (r ≥ 1), U periodic complementary domain, Then,

1 no periodic points in ∂ U, 2 ∂ U = finite disjoint union of aperiodic annular continua.

Example: p ∈ Perh(f ), U = connected component of S \ W s(p).

Remarks

1 Mather [1981] proved ρ /

∈ Q, assuming r large (r ≥ 16, KAM & KS)

2 For S2 and T2, can be proved using Mather + Pixton-Oliveira. 3 Generic condition is explicit. Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Theorem C’

f a C r-generic area preserving diffeomorphism of a closed surface (r ≥ 1) U periodic open set with finitely many topological ends. Then ∂ U = {aperiodic annular continua} ⊔ {periodic points} (finitely many of each)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Corollary C’ completes the proof of:

Theorem D

For a C r-generic area-preserving diffeo f of any closed surface,

  • p∈Per(f )

W s(p) =

  • p∈Per(f )

W u(p) = S

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Application for generic area-preserving diffeos

Corollary C’ completes the proof of:

Theorem D

For a C r-generic area-preserving diffeo f of any closed surface,

  • p∈Per(f )

W s(p) =

  • p∈Per(f )

W u(p) = S For S = S2, r ≥ 16: done by Franks and Le Calvez [ETDS, 2003] For any genus: proof of J. Xia [CMP, 2006] relies in Corollary C’ (gap).

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime ends

Definition

cross-cut: a simple arc γ in U with endpoints in ∂ U. cross-section: any one of the two components of U \ γ.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Chains

A chain in U is a decreasing sequence of cross sections (Dn) bounded by cross-cuts (γn) such that γn ∩ γn+1 = ∅.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime ends

A chain in U is a decreasing sequence of cross sections (Dn) bounded by cross-cuts (γn) such that γn ∩ γn+1 = ∅. If (D′

n) is another chain, we say that (Dn) divides (D′ n) if for each n > 0

there is m such that Dm ⊂ D′

n.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime ends

A chain in U is a decreasing sequence of cross sections (Dn) bounded by cross-cuts (γn) such that γn ∩ γn+1 = ∅. If (D′

n) is another chain, we say that (Dn) divides (D′ n) if for each n > 0

there is m such that Dm ⊂ D′

n.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime ends

A chain in U is a decreasing sequence of cross sections (Dn) bounded by cross-cuts (γn) such that γn ∩ γn+1 = ∅. If (D′

n) is another chain, we say that (Dn) divides (D′ n) if for each n > 0

there is m such that Dm ⊂ D′

n.

A chain (Dn) is called a prime chain if it divides (D′

n)∈N whenever (D′ n) is

a chain that divides (Dn).

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime ends

A chain in U is a decreasing sequence of cross sections (Dn) bounded by cross-cuts (γn) such that γn ∩ γn+1 = ∅. If (D′

n) is another chain, we say that (Dn) divides (D′ n) if for each n > 0

there is m such that Dm ⊂ D′

n.

A chain (Dn) is called a prime chain if it divides (D′

n)∈N whenever (D′ n) is

a chain that divides (Dn). Prime ends of U = PE(U) := {prime chains} / equivalence

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime chain

If U is compact, then we may define in this way: A prime chain in U is a decreasing sequence of cross sections (Dn) bounded by cross-cuts (γn) such that diam(γn) → 0 as n → ∞ γn ∩ γn+1 = ∅

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Introduction Area preserving surface diffeomorphism

Prime ends

Prime ends compactification (Carath´ eodory)

PE(U) ≃ S1

  • U := U ⊔ PE(U) ≃ D

Prime ends rotation number

f extends to a homeomorphism f : U → U ρ(f , U) = Poincar´ e rotation number of

  • f |S1

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

N-translation arc

γ simple arc from x to f (x) = x, Γ = γ ∪ f (γ) ∪ · · · ∪ f N(γ) is also a simple arc.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

N-translation arc

γ simple arc from x to f (x) = x, Γ = γ ∪ f (γ) ∪ · · · ∪ f N(γ) is also a simple arc.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Theorem E (Arc Lemma)

f : S → S homeomorphism, S surface of genus g. U ⊂ S invariant open topological disk (S \ U one point) f is nonwandering in U, ρ(f , U) = α = 0 = ⇒ ∃N = Nα,g s.t every N-translation arc in S is disjoint from ∂ U.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Theorem E (Arc Lemma)

f : S → S homeomorphism, S surface of genus g. U ⊂ S invariant open topological disk (S \ U one point) f is nonwandering in U, ρ(f , U) = α = 0 = ⇒ ∃N = Nα,g s.t every N-translation arc in S is disjoint from ∂ U. Case N = ∞ and ”transverse”: easy.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Theorem E (Arc Lemma)

f : S → S homeomorphism, S surface of genus g. U ⊂ S invariant open topological disk (S \ U one point) f is nonwandering in U, ρ(f , U) = α = 0 = ⇒ ∃N = Nα,g s.t every N-translation arc in S is disjoint from ∂ U. Case N = ∞ and ”transverse”: easy. ”non-transverse” (⊂ ∂ U): hard.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Theorem E (Arc Lemma)

f : S → S homeomorphism, S surface of genus g. U ⊂ S invariant open topological disk (S \ U one point) f is nonwandering in U, ρ(f , U) = α = 0 = ⇒ ∃N = Nα,g s.t every N-translation arc in S is disjoint from ∂ U. Case N = ∞ and ”transverse”: easy. ”non-transverse” (⊂ ∂ U): hard.

Remark (Brouwer theory)

Assuming S = R2: Every non-fixed point belongs to an 1-translation arc γ. If γ is not an N-translation arc, then Γ = γ ∪ f (γ) ∪ · · · ∪ f N(γ) surrounds a fixed point.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Arc Lemma: idea of the proof

Assume S = R2. In this case, N = 3

Let γ be a 3-translation arc intersecting ∂ U ∃ maximal cross-cut γ0 defined by γ Cyclic order of iterations of γ0 by rotation number Linear order of iterations of γ0 by 3-translation arc γ Construct a pair of simple closed curves with intersection number = 1 = ⇒ genus of S > 0. Contradiction !

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Arc Lemma: idea of the proof (heuristics)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Arc Lemma: idea of the proof (heuristics)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Arc Lemma: idea of the proof (heuristics)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Arc Lemma: idea of the proof (heuristics)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Idea of the proof of Theorem A

f : R2 → R2 homeomorphism U ⊂ R2 , simply connected, open, f -invariant f is nonwandering in U

Theorem A (Converse of [C-L])

ρ(f , U) = 0 = ⇒ ∄ fixed point in ∂ U

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Idea of the proof of Theorem A

f : R2 → R2 homeomorphism U ⊂ R2 , simply connected, open, f -invariant f is nonwandering in U

Theorem A (Converse of [C-L])

ρ(f , U) = 0 = ⇒ ∄ fixed point in ∂ U

General strategy

Assuming there is a fixed point z1 in ∂ U: Find an N-translation arc in a neighborhood of z1 contradicts Arc Lemma.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Idea of the proof of Theorem A

f : R2 → R2 homeomorphism U ⊂ R2 , simply connected, open, f -invariant f is nonwandering in U

Theorem A (Converse of [C-L])

ρ(f , U) = 0 = ⇒ ∄ fixed point in ∂ U

General strategy

Assuming there is a fixed point z1 in ∂ U: Find an N-translation arc in a neighborhood of z1 contradicts Arc Lemma. Problem: doesn’t work directly!

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Idea of the proof of Theorem A

Suppose ρ(f , U) = 0 but z1 ∈ Fix(f ) ∩ ∂ U

Reduce to the case where:

∃ unique fixed point z0 ∈ U. ∄ accessible fixed point in ∂ U Fix(f ) totally disconnected.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Idea of the proof of Theorem A

Suppose ρ(f , U) = 0 but z1 ∈ Fix(f ) ∩ ∂ U

Reduce to the case where:

∃ unique fixed point z0 ∈ U. ∄ accessible fixed point in ∂ U Fix(f ) totally disconnected. Remove maximal unlinked X ⊂ Fix(f ), z0, z1 ∈ X. (using the work of O. Jaulent [2012])

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

Idea of the proof of Theorem A

Suppose ρ(f , U) = 0 but z1 ∈ Fix(f ) ∩ ∂ U

Reduce to the case where:

∃ unique fixed point z0 ∈ U. ∄ accessible fixed point in ∂ U Fix(f ) totally disconnected. Remove maximal unlinked X ⊂ Fix(f ), z0, z1 ∈ X. (using the work of O. Jaulent [2012]) M = R2 \ (X \ {z0}), π : M → M universal covering map Define f : M → M, U invariant for f , same rotation number. π(Fix( f )) far from z1 Find N-translation arc γ for f that projects near z1. Brouwer = ⇒ Γ “turns around” a fixed point of f . Contradiction!

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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Ideas of proofs

a technical problem

In the classic theory of prime ends: U must be a bounded (relatively compact) open subset of S PE compactification depends fundamentally on ambient space = ⇒ so does prime ends dynamics. Rotation number?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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SLIDE 70

Ideas of proofs

a technical problem

In the classic theory of prime ends: U must be a bounded (relatively compact) open subset of S PE compactification depends fundamentally on ambient space = ⇒ so does prime ends dynamics. Rotation number?

Theorem

  • The theory of prime ends extends to the unbounded case;
  • If U ⊂ S′ ⊂ S open invariant sets and ∂S′ U = ∅, then

ρ(f , U ⊂ S) = ρ(f , U ⊂ S′)

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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SLIDE 71

Ideas of proofs

Further problems and results

Poincar´ e theory on S1

Rotation number is independent of the point used to compute it. ρ(f ) = p/q ∈ Q = ⇒ Fix(f q) = ∅ ρ(f ) / ∈ Q = ⇒ Per(f ) = ∅.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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SLIDE 72

Ideas of proofs

Further problems and results

Poincar´ e theory on S1

Rotation number is independent of the point used to compute it. ρ(f ) = p/q ∈ Q = ⇒ Fix(f q) = ∅ ρ(f ) / ∈ Q = ⇒ Per(f ) = ∅. Cartwright-Littlewod + Theorem A = ⇒ this holds for boundary dynamics (+nonwandering).

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

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SLIDE 73

Ideas of proofs

Further problems and results

Refinement of Poincar´ e theory on S1

ρ(f ) = p/q ∈ Q = ⇒ Fix(f q) = ∅ and α(x) ∪ ω(x) ⊂ Fix(f q) for all x ∈ S1. ρ(f ) / ∈ Q = ⇒ Per(f ) = ∅, there is a unique minimal set, and f is uniquely ergodic.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

slide-74
SLIDE 74

Ideas of proofs

Further problems and results

Refinement of Poincar´ e theory on S1

ρ(f ) = p/q ∈ Q = ⇒ Fix(f q) = ∅ and α(x) ∪ ω(x) ⊂ Fix(f q) for all x ∈ S1. ρ(f ) / ∈ Q = ⇒ Per(f ) = ∅, there is a unique minimal set, and f is uniquely ergodic. How much of this translates to boundary dynamics?

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014

slide-75
SLIDE 75

Ideas of proofs

Further problems and results

Refinement of Poincar´ e theory on S1

ρ(f ) = p/q ∈ Q = ⇒ Fix(f q) = ∅ and α(x) ∪ ω(x) ⊂ Fix(f q) for all x ∈ S1. ρ(f ) / ∈ Q = ⇒ Per(f ) = ∅, there is a unique minimal set, and f is uniquely ergodic. How much of this translates to boundary dynamics?

Work in progress

The first item holds for boundary dynamics with a nonwandering condition.

Meysam Nassiri (IPM) Boundary dynamics and topology Surfaces at SP, 2014