A semigroup with identity generated (w.r.t. the composition) by a family of diffeomorphisms Φ = { φ 1 , . . . , φ k } on S 1 , Iterated Function Systems = { h : S 1 → S 1 : h = φ i n ◦ · · · ◦ φ i 1 , i j ∈ { 1 , . . . , k }} ∪ { id } def IFS (Φ) on the circle is called iterated function system or shortly IFS. Pablo G. Barrientos and Artem Raibekas Universidad de Oviedo (Spain) Universidade Federal Fluminense (Brasil) ICDEA: 27 July 2012 Let Λ ⊂ S 1 . We say that Λ is A semigroup with identity generated (w.r.t. the composition) - invariant for IFS (Φ) if Orb Φ ( x ) ⊂ Λ for all x ∈ Λ , by a family of diffeomorphisms Φ = { φ 1 , . . . , φ k } on S 1 , - minimal for IFS (Φ) if = { h : S 1 → S 1 : h = φ i n ◦ · · · ◦ φ i 1 , i j ∈ { 1 , . . . , k }} ∪ { id } def IFS (Φ) for all x ∈ Λ . Λ ⊂ Orb Φ ( x ) 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 Let Λ ⊂ S 1 . We say that Λ is - invariant for IFS (Φ) if Orb Φ ( x ) ⊂ Λ for all x ∈ Λ , - invariant for IFS (Φ) if Orb Φ ( x ) ⊂ Λ for all x ∈ Λ , - minimal for IFS (Φ) if - minimal for IFS (Φ) if Λ ⊂ Orb Φ ( x ) for all x ∈ Λ . Λ ⊂ Orb Φ ( x ) for all x ∈ Λ . In order to define robust properties under perturbations we In order to define robust properties under perturbations we introduce the following concept of proximity into the set of introduce the following concept of proximity into the set of IFSs. We say that IFSs. We say that IFS ( ψ 1 , . . . , ψ k ) is C r -close to IFS ( φ 1 , . . . , φ k ) 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 . 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 Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: f : S 1 → S 1 we have three possibilities: has a periodic orbit, - f - IFS ( f ) has a finite orbit, - all the orbits (for forward iterates) of f are dense, - all the orbits (for forward iterates) of f are dense, - there is a wandering interval for f . - there is a wandering interval for f . The wandering intervals are the gaps of a unique f -invariant The wandering intervals are the gaps of a unique f -invariant minimal Cantor set Λ ⊂ S 1 . minimal Cantor set Λ ⊂ S 1 .
Taking into account the rotation number of a homeomorphism Taking into account the rotation number of a homeomorphism f : S 1 → S 1 we have three possibilities: f : S 1 → S 1 we have three possibilities: - IFS ( f ) has a finite orbit, - IFS ( f ) has a finite orbit, - S 1 is minimal for IFS ( f ) , - S 1 is minimal for IFS ( f ) , - there exists an invariant minimal Cantor set for IFS ( f ) . - there is a wandering interval for f . In this case it is unique. 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 THEOREM ( Denjoy ): There exists such that if ε > 0 f : S 1 → S 1 we have three possibilities: 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 ) . - IFS ( f ) has a finite orbit, - S 1 is minimal for IFS ( f ) , Moreover, the following conditions are equivalent: 1. S 1 is minimal for IFS ( f ) , - there exists an invariant minimal Cantor set for IFS ( f ) . 2. there are no periodic points for 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.
ss - intervals for IFS (Φ) 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 Given Φ = { f 0 , f 1 } ⊂ Diff 1 then there are no invariant minimal Cantor sets for IFS ( f ) . DEFINITION: + ( R ) , an interval [ p 0 , p 1 ] ⊂ R is called ss -interval for IFS (Φ) if: Moreover, the following conditions are equivalent: - [ p 0 , p 1 ] = f 0 ([ p 0 , p 1 ]) ∪ f 1 ([ p 0 , p 1 ]) , 1. S 1 is minimal for IFS ( f ) , - ( p 0 , p 1 ) ∩ Fix ( f i ) � = ∅ for i = 1 , 2 , and p j �∈ Fix ( f i ) for i � = j , 2. there are no periodic points for f . - 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) . 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 ) . 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 ) . f 0 a Condition (2) is satisfied if f 0 and f 1 have not periodic points in common. p 0 p 1 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 Improved Duminy ’ s Lemma 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, Let K ss THEOREM: be a ss -interval for IFS (Φ) with Φ 3. Λ is a Cantor set. 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 ) . Let x ∈ S 1 . The ω -limit of x for IFS (Φ) is the set Then exists a non-empty closed set Λ ⊂ S 1 such that = { y ∈ S 1 : ∃ ( h n ) n ⊂ IFS (Φ) \{ id } s.t. def n →∞ h n ◦· · ·◦ h 1 ( x ) = y } , ω Φ ( x ) lim Λ = φ 1 (Λ) ∪ · · · ∪ φ k (Λ) = Orb Φ ( x ) for all x ∈ Λ . One (and only one) of the following possibilities occurs: while the ω -limit of IFS (Φ) is 1. Λ is a finite orbit, ∃ x ∈ S 1 s.t. y ∈ ω Φ ( x ) } { y ∈ S 1 : def � � ω ( IFS (Φ)) = cl , 2. Λ has non-empty interior, 3. Λ is a Cantor set. where " cl " denotes the closure of a set. Similarly we define the α -limit of IFS (Φ) . Finally, the limit set of IFS (Φ) Denjoy ’ s Theorem for IFS L ( IFS (Φ)) = ω ( IFS (Φ)) ∪ α ( IFS (Φ)) . 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. Spectral decomposition for IFS 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 ) } THEOREM: There exists ε > 0 such that if f 0 , f 1 ∈ Diff 2 ( S 1 ) { y ∈ S 1 : def � � ω ( IFS (Φ)) = cl , are Morse-Smale diffeomorphisms of periods n 0 and n 1 , where " cl " denotes the closure of a set. Similarly we define respectively, ε -close to the identity in the C 2 -topology and with no periodic point in common, then there are the α -limit of IFS (Φ) . Finally, the limit set of IFS (Φ) finitely many isolated, transitive pairwise disjoint intervals L ( IFS (Φ)) = ω ( IFS (Φ)) ∪ α ( IFS (Φ)) . K 1 , . . . , K m for IFS ( f n 0 0 , f n 1 1 ) such that Let Λ ⊂ S 1 . We say that Λ is m � L ( IFS ( f n 0 0 , f n 1 1 )) = K i . - transitive for IFS (Φ) if there exists a dense orbit in Λ , i = 1 - isolated for IFS (Φ) if Λ ∩ Per ( IFS (Φ)) � = ∅ and there exists Moreover, this decomposition is C 1 -robust. an open set D such that and Λ ⊂ D Per ( IFS (Φ)) ∩ D ⊂ Λ .
Thanks for your attention
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