Iterated Function Systems on the circle Pablo G. Barrientos and - PowerPoint PPT Presentation

Iterated Function Systems on the circle Pablo G. Barrientos and Artem Raibekas Universidad de Oviedo (Spain) Universidade Federal Fluminense (Brasil) ICDEA: 27 July 2012

A semigroup with identity generated (w.r.t. the composition) by a family of diffeomorphisms Φ = { φ 1 , . . . , φ k } on S 1 , = { h : S 1 → S 1 : h = φ i n ◦ · · · ◦ φ i 1 , i j ∈ { 1 , . . . , k }} ∪ { id } def IFS (Φ) is called iterated function system or shortly IFS.

A semigroup with identity generated (w.r.t. the composition) by a family of diffeomorphisms Φ = { φ 1 , . . . , φ k } on S 1 , = { h : S 1 → S 1 : h = φ i n ◦ · · · ◦ φ i 1 , i j ∈ { 1 , . . . , k }} ∪ { id } def IFS (Φ) is called iterated function system or shortly IFS. For each x ∈ S 1 , we define the orbit of x for IFS (Φ) as def = { h ( x ): h ∈ IFS (Φ) } ⊂ S 1 Orb Φ ( x ) and the set of periodic points of IFS (Φ) as = { x ∈ S 1 : def h ( x ) = x for some h ∈ IFS (Φ) , h � = id } . Per ( IFS (Φ))

Let Λ ⊂ S 1 . We say that Λ is - invariant for IFS (Φ) if Orb Φ ( x ) ⊂ Λ for all x ∈ Λ , - minimal for IFS (Φ) if for all x ∈ Λ . Λ ⊂ Orb Φ ( x )

Let Λ ⊂ S 1 . We say that Λ is - invariant for IFS (Φ) if Orb Φ ( x ) ⊂ Λ for all x ∈ Λ , - minimal for IFS (Φ) if for all x ∈ Λ . Λ ⊂ Orb Φ ( x ) In order to define robust properties under perturbations we introduce the following concept of proximity into the set of IFSs. We say that IFS ( ψ 1 , . . . , ψ k ) is C r -close to IFS ( φ 1 , . . . , φ k ) if ψ i is C r -close to φ i for all i = 1 , . . . , k .

Let Λ ⊂ S 1 . We say that Λ is - invariant for IFS (Φ) if Orb Φ ( x ) ⊂ Λ for all x ∈ Λ , - minimal for IFS (Φ) if for all x ∈ Λ . Λ ⊂ Orb Φ ( x ) In order to define robust properties under perturbations we introduce the following concept of proximity into the set of IFSs. We say that IFS ( ψ 1 , . . . , ψ k ) is C r -close to IFS ( φ 1 , . . . , φ k ) if ψ i is C r -close to φ i for all i = 1 , . . . , k . So, we will say that S 1 is C r -robust minimal for IFS (Φ) if S 1 is minimal for all IFS (Ψ) C r -close enough to IFS (Φ) .

Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: has a periodic orbit, - f - all the orbits (for forward iterates) of f are dense, - there is a wandering interval for f . The wandering intervals are the gaps of a unique f -invariant minimal Cantor set Λ ⊂ S 1 .

Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: - IFS ( f ) has a finite orbit, - all the orbits (for forward iterates) of f are dense, - there is a wandering interval for f . The wandering intervals are the gaps of a unique f -invariant minimal Cantor set Λ ⊂ S 1 .

Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: - IFS ( f ) has a finite orbit, - S 1 is minimal for IFS ( f ) , - there is a wandering interval for f . The wandering intervals are the gaps of a unique f -invariant minimal Cantor set Λ ⊂ S 1 .

Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: - IFS ( f ) has a finite orbit, - S 1 is minimal for IFS ( f ) , - there exists an invariant minimal Cantor set for IFS ( f ) . In this case it is unique.

Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: - IFS ( f ) has a finite orbit, - S 1 is minimal for IFS ( f ) , - there exists an invariant minimal Cantor set for IFS ( f ) . In this case it is unique. This trichotomy can be extended to actions of groups of homeomorphisms on the circle: THEOREM ( Ghys ): Let G (Φ) be a subgroup of Hom ( S 1 ) . Then one (and only one) possibility occurs: - G (Φ) has a finite orbit, - S 1 is minimal for G (Φ) , - there exists an invariant minimal Cantor set for G (Φ) . In this case it is unique.

THEOREM ( Denjoy ): There exists ε > 0 such that if f ∈ Diff 2 ( S 1 ) is ε -close to the identity in the C 2 -topology then there are no invariant minimal Cantor sets for IFS ( f ) . Moreover, the following conditions are equivalent: 1. S 1 is minimal for IFS ( f ) , 2. there are no periodic points for f .

THEOREM ( Denjoy ): There exists ε > 0 such that if f ∈ Diff 2 ( S 1 ) is ε -close to the identity in the C 2 -topology then there are no invariant minimal Cantor sets for IFS ( f ) . Moreover, the following conditions are equivalent: 1. S 1 is minimal for IFS ( f ) , 2. there are no periodic points for f . THEOREM ( Generalized Duminy ): There exists ε > 0 such that if f 0 , f 1 ∈ Diff 2 ( S 1 ) are Morse-Smale ε -close to the identity in the C 2 -topology then there are no invariant minimal Cantor sets for all G (Ψ) C 1 -close to G ( f 0 , f 1 ) . Moreover, the following conditions are equivalent a : 1. S 1 is C 1 -robust minimal for G ( f 0 , f 1 ) , 2. f 1 ( Per ( f 0 )) � = Per ( f 0 ) . a Condition (2) is satisfied if f 0 and f 1 have not periodic points in common.

ss - intervals for IFS (Φ) Given Φ = { f 0 , f 1 } ⊂ Diff 1 DEFINITION: + ( R ) , an interval [ p 0 , p 1 ] ⊂ R is called ss -interval for IFS (Φ) if: - [ p 0 , p 1 ] = f 0 ([ p 0 , p 1 ]) ∪ f 1 ([ p 0 , p 1 ]) , - ( p 0 , p 1 ) ∩ Fix ( f i ) � = ∅ for i = 1 , 2 , and p j �∈ Fix ( f i ) for i � = j , - p 0 and p 1 are attracting fixed points of f 0 and f 1 resp. We will denote by K ss Φ a ss -interval [ p 0 , p 1 ] for IFS (Φ) . f 1 f 0 p 0 p 1

Improved Duminy ’ s Lemma Let K ss THEOREM: be a ss -interval for IFS (Φ) with Φ Φ = { f 0 , f 1 } ⊂ Diff 2 + ( R ) such that f i | K ss has hyperbolic fixed Φ points. Then, there exists ε ≥ 0 . 16 such that if f 0 | K ss Φ , f 1 | K ss Φ are ε -close to the identity in the C 2 -topology, it holds K ss K ss for all x ∈ K ss Ψ ⊂ Per ( IFS (Ψ)) and Ψ = Orb Ψ ( x ) Ψ , for every IFS (Ψ) C 1 -close to IFS (Φ) .

THEOREM: Consider IFS (Φ) with Φ = { φ 1 , . . . , φ k } ⊂ Hom ( S 1 ) . Then exists a non-empty closed set Λ ⊂ S 1 such that Λ = φ 1 (Λ) ∪ · · · ∪ φ k (Λ) = Orb Φ ( x ) for all x ∈ Λ . One (and only one) of the following possibilities occurs: 1. Λ is a finite orbit, 2. Λ has non-empty interior, 3. Λ is a Cantor set.

THEOREM: Consider IFS (Φ) with Φ = { φ 1 , . . . , φ k } ⊂ Hom ( S 1 ) . Then exists a non-empty closed set Λ ⊂ S 1 such that Λ = φ 1 (Λ) ∪ · · · ∪ φ k (Λ) = Orb Φ ( x ) for all x ∈ Λ . One (and only one) of the following possibilities occurs: 1. Λ is a finite orbit, 2. Λ has non-empty interior, 3. Λ is a Cantor set. Denjoy ’ s Theorem for IFS if f 0 , f 1 ∈ Diff 2 ( S 1 ) are THEOREM: There exists ε > 0 s.t. Morse-Smale diff. ε -close to the identity in the C 2 -topology with no periodic point in common then, there are no invari- ant minimal Cantor sets for all IFS (Ψ) C 1 -close to IFS ( f 0 , f 1 ) . Moreover, denoting by n i the period of f i , it is equivalent: 1. S 1 is C 1 -robust minimal for IFS ( f n 0 0 , f n 1 1 ) , 2. there are no ss -intervals for IFS ( f n 0 0 , f n 1 1 ) .

Let x ∈ S 1 . The ω -limit of x for IFS (Φ) is the set = { y ∈ S 1 : ∃ ( h n ) n ⊂ IFS (Φ) \{ id } s.t. def ω Φ ( x ) n →∞ h n ◦· · ·◦ h 1 ( x ) = y } , lim while the ω -limit of IFS (Φ) is ∃ x ∈ S 1 s.t. y ∈ ω Φ ( x ) } { y ∈ S 1 : def � � ω ( IFS (Φ)) = cl , where " cl " denotes the closure of a set. Similarly we define the α -limit of IFS (Φ) . Finally, the limit set of IFS (Φ) L ( IFS (Φ)) = ω ( IFS (Φ)) ∪ α ( IFS (Φ)) .

Let x ∈ S 1 . The ω -limit of x for IFS (Φ) is the set = { y ∈ S 1 : ∃ ( h n ) n ⊂ IFS (Φ) \{ id } s.t. def ω Φ ( x ) n →∞ h n ◦· · ·◦ h 1 ( x ) = y } , lim while the ω -limit of IFS (Φ) is ∃ x ∈ S 1 s.t. y ∈ ω Φ ( x ) } { y ∈ S 1 : def � � ω ( IFS (Φ)) = cl , where " cl " denotes the closure of a set. Similarly we define the α -limit of IFS (Φ) . Finally, the limit set of IFS (Φ) L ( IFS (Φ)) = ω ( IFS (Φ)) ∪ α ( IFS (Φ)) . Let Λ ⊂ S 1 . We say that Λ is - transitive for IFS (Φ) if there exists a dense orbit in Λ , - isolated for IFS (Φ) if Λ ∩ Per ( IFS (Φ)) � = ∅ and there exists an open set D such that and Λ ⊂ D Per ( IFS (Φ)) ∩ D ⊂ Λ .

Spectral decomposition for IFS THEOREM: There exists ε > 0 such that if f 0 , f 1 ∈ Diff 2 ( S 1 ) are Morse-Smale diffeomorphisms of periods n 0 and n 1 , respectively, ε -close to the identity in the C 2 -topology and with no periodic point in common, then there are finitely many isolated, transitive pairwise disjoint intervals K 1 , . . . , K m for IFS ( f n 0 0 , f n 1 1 ) such that m L ( IFS ( f n 0 0 , f n 1 � 1 )) = K i . i = 1 Moreover, this decomposition is C 1 -robust.

Thanks for your attention