Skeletons for transitive fibered maps Katrin Gelfert (UFRJ, Brazil) - PowerPoint PPT Presentation

Skeletons for transitive fibered maps Katrin Gelfert (UFRJ, Brazil) joint work with L. J. Díaz and M. Rams March 29, 2016 Skeletons for transitive fibered maps March 29, 2016 1 / 10

Axioms: Some notation Consider a one step skew-product F : Σ k × S 1 → Σ k × S 1 � � F ( ξ, x ) = σ ( ξ ) , f ξ 0 ( x ) . Consider the associated IFS { f i } k − 1 i = 0 . Some notation: Given finite sequences ( ξ 0 . . . ξ n ) and ( ξ − m . . . ξ − 1 ) , let def f [ ξ 0 ... ξ n ] = f ξ n ◦ · · · ◦ f ξ 1 ◦ f ξ 0 = ( f ξ − 1 ◦ . . . ◦ f ξ − m ) − 1 = ( f [ ξ − m ... ξ − 1 ] ) − 1 def f [ ξ − m ... ξ − 1 . ] Given A ⊂ S 1 , define its forward and backward orbit , respectively, by � � O + ( A ) def = f [ β 0 ... β n − 1 ] ( A ) n ≥ 0 ( β 0 ...β n − 1 ) O − ( A ) def � � = f [ θ − m ... θ − 1 . ] ( A ) m ≥ 1 ( θ − m ...θ − 1 ) Skeletons for transitive fibered maps March 29, 2016 2 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T CEC+( J ) CEC − ( J ) Acc + ( J ) Acc − ( J ) Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). CEC+( J ) CEC − ( J ) Acc + ( J ) Acc − ( J ) Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) Acc + ( J ) Acc − ( J ) Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) (CE backward Covering). Acc + ( J ) Acc − ( J ) Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) (CE backward Covering). Acc + ( J ) (Forward Accessibility). Acc − ( J ) Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) (CE backward Covering). Acc + ( J ) (Forward Accessibility). Acc − ( J ) (Backward Accessibility). Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). ∃ x ∈ S 1 : O + ( x ) and O − ( x ) are both dense in S 1 . CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) (CE backward Covering). Acc + ( J ) (Forward Accessibility). Acc − ( J ) (Backward Accessibility). Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). ∃ x ∈ S 1 : O + ( x ) and O − ( x ) are both dense in S 1 . CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) (CE backward Covering). Acc + ( J ) (Forward Accessibility). O + ( int J ) = S 1 . Acc − ( J ) (Backward Accessibility). Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). ∃ x ∈ S 1 : O + ( x ) and O − ( x ) are both dense in S 1 . CEC+( J ) (Controlled Expanding forward Covering). CEC − ( J ) (CE backward Covering). Acc + ( J ) (Forward Accessibility). O + ( int J ) = S 1 . Acc − ( J ) (Backward Accessibility). O − ( int J ) = S 1 . Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). ∃ x ∈ S 1 : O + ( x ) and O − ( x ) are both dense in S 1 . CEC+( J ) (Controlled Expanding forward Covering). ∃ K 1 , . . . , K 5 : for every interval H ⊂ S 1 intersecting J with | H | < K 1 ∃ ( η 0 . . . η ℓ − 1 ) , ℓ ≤ K 2 | log | H || + K 3 , such that f [ η 0 ... η ℓ − 1 ] ( H ) ⊃ B ( J , K 4 ) , ∀ x ∈ H � ≥ ℓ K 5 , � ( f [ η 0 ... η ℓ − 1 ] ) ′ ( x ) � � log K 5 > 1 . CEC − ( J ) (CE backward Covering). Acc + ( J ) (Forward Accessibility). O + ( int J ) = S 1 . Acc − ( J ) (Backward Accessibility). O − ( int J ) = S 1 . Skeletons for transitive fibered maps March 29, 2016 3 / 10

Axioms: Let J ⊂ S 1 be a closed blending interval . T (Transitivity). ∃ x ∈ S 1 : O + ( x ) and O − ( x ) are both dense in S 1 . CEC+( J ) (Controlled Expanding forward Covering). ∃ K 1 , . . . , K 5 : for every interval H ⊂ S 1 intersecting J with | H | < K 1 ∃ ( η 0 . . . η ℓ − 1 ) , ℓ ≤ K 2 | log | H || + K 3 , such that f [ η 0 ... η ℓ − 1 ] ( H ) ⊃ B ( J , K 4 ) , ∀ x ∈ H � ≥ ℓ K 5 , � � ( f [ η 0 ... η ℓ − 1 ] ) ′ ( x ) � log K 5 > 1 . CEC − ( J ) (CE backward Covering). IFS { f − 1 } i satisfies CEC +( J ). i Acc + ( J ) (Forward Accessibility). O + ( int J ) = S 1 . Acc − ( J ) (Backward Accessibility). O − ( int J ) = S 1 . Skeletons for transitive fibered maps March 29, 2016 3 / 10

Examples. System that satisfies Axioms T, CEC ± , Acc ± One-dimensional blenders Motivated by: [Bonatti, Díaz ’96], [Bonatti, Díaz, Ures ’02] IFS { f i } k − 1 i = 0 , k ≥ 2, has expanding blender if: there are [ c , d ] ⊂ [ a , b ] ⊂ S 1 so that (expansion) f ′ 0 ( x ) ≥ β > 1 ∀ x ∈ [ a , b ] f 0 (boundary condition) f 0 ( a ) = f 1 ( c ) = a (covering and invariance) f 0 ([ a , d ]) = [ a , b ] and f 1 ([ c , b ]) ⊂ [ a , b ] f 1 It has a contracting blender if { f − 1 } i does. i Suppose that ∀ x ∈ S 1 by some forward iteration maps inside an expanding blender a c d b ( a , b ) and by some backward iteration meets a contracting blender. Skeletons for transitive fibered maps March 29, 2016 4 / 10

Examples. System that satisfies Axioms T, CEC ± , Acc ± Contraction-expansion-rotation examples Motivated by: [Gorodetskii, Il’yashenko, Kleptsyn, Nal’skii ’05] Consider IFS { f i } k − 1 i = 0 , k ≥ 3, so that f 0 has a repelling fixed point, f 0 f 1 f 2 f 1 has an attracting fixed point, f 2 is an irrational rotation. Skeletons for transitive fibered maps March 29, 2016 5 / 10

Main results Theorem (Approximating non-hyperbolic measure by hyperbolic ones) Let µ ∈ M erg with χ ( µ ) = 0 and h = h ( µ ) > 0 . Then ∀ γ, δ, λ > 0 there exists compact F-invariant transitive hyperbolic Γ + h top (Γ + ) ≥ h ( µ ) − γ and for every ν ∈ M erg (Γ + ) d w ∗ ( ν, µ ) < δ and χ ( ν ) ∈ ( 0 , λ ) . Analogously with hyperbolic Γ − with χ ( ν ) ∈ ( − λ, 0 ) for ν ∈ M erg (Γ − ) . Skeletons for transitive fibered maps March 29, 2016 6 / 10

Main results Theorem (Approximating non-hyperbolic measure by hyperbolic ones) Let µ ∈ M erg with χ ( µ ) = 0 and h = h ( µ ) > 0 . Then ∀ γ, δ, λ > 0 there exists compact F-invariant transitive hyperbolic Γ + h top (Γ + ) ≥ h ( µ ) − γ and for every ν ∈ M erg (Γ + ) d w ∗ ( ν, µ ) < δ and χ ( ν ) ∈ ( 0 , λ ) . Analogously with hyperbolic Γ − with χ ( ν ) ∈ ( − λ, 0 ) for ν ∈ M erg (Γ − ) . Theorem (Restricted variational principle for entropy) h top ( F ) = sup h ( µ ) = sup h ( µ ) ≥ sup h ( µ ) . µ ∈M erg ,< 0 µ ∈M erg ,> 0 µ ∈M erg , = 0 Skeletons for transitive fibered maps March 29, 2016 6 / 10

Main results Theorem (“Perturbing” hyperbolic measure “toward the other side”) Let µ ∈ M erg with α = χ ( µ ) < 0 and h = h ( µ ) > 0 . Then ∀ γ, δ > 0 , ∀ β > 0 exists compact F-invariant transitive hyperbolic Γ h h top (Γ) ≥ 1 + K 2 ( β + | α | ) − γ and for every ν ∈ M erg (Γ) β β 1 + K 2 ( β + | α | ) − δ < χ ( ν ) < + δ, 1 1 + log � F � ( β + | α | ) 1 d w ∗ ( ν, µ ) < 1 − 1 + K 2 ( β + | α | ) + δ Here � � def def | f ′ i ( x ) | , | ( f − 1 ) ′ ( x ) | K 2 = inf { K 2 ( J ): J is blending interval } , � F � = max . i i , x Analogous result is true for µ with χ ( µ ) = α > 0. Skeletons for transitive fibered maps March 29, 2016 7 / 10

Ingredients: Skeletons F has the skeleton property relative to J ⊂ S 1 , h ≥ 0, α ≥ 0 if: There exist connecting times m b , m f ∈ N : ∀ ε H ∈ ( 0 , h ) ∀ ε E > 0 ∃ n 0 ≥ 1 so that ∀ m ≥ n 0 there exists a finite set X = X ( h , α, ε H , ε E , m ) = { X i } of points X i = ( ξ i , x i ) : (i) card X ≍ e m ( h ± ε H ) , (ii) the sequences ( ξ i 0 . . . ξ i m − 1 ) are all different, (iii) 1 n − 1 ] ) ′ ( x i ) | ≍ α ± ε E ∀ n = 0 , . . . , m . n log | ( f [ ξ i 0 ... ξ i ∃ sequences ( θ i 1 . . . θ i r i ) , r i ≤ m f , ( β i 1 . . . β i s i ) , s i ≤ m b , points x ′ i ∈ J : ri ] ( x ′ (iv) f [ θ i i ) = x i , 1 ... θ i (v) f [ ξ i si ] ( x i ) ∈ J . 0 ... ξ i m − 1 β i 1 ... β i Skeletons for transitive fibered maps March 29, 2016 8 / 10

Ingredients: Multi-variable-time horseshoes Let T : X → X be a local homeomorphism of a compact metric space. { S i } M i = 1 disjoint compact, t ij ∈ { t min , . . . , t max } transition times : T t ij ( S i ) ⊃ S j , T t ij | S i ∩ T − tij ( S j ) injective . Σ + T t 11 M S 1 S 13 T t 13 S 11 S 12 S 3 T t 12 S 2 S 1 Skeletons for transitive fibered maps March 29, 2016 9 / 10