Dynamical systems Expanding maps on the circle. Semiconjugacy Jana - PowerPoint PPT Presentation

coding E 2 is a factor of σ Expanding maps are factors of σ Dynamical systems Expanding maps on the circle. Semiconjugacy Jana Rodriguez Hertz ICTP 2018

coding E 2 is a factor of σ Expanding maps are factors of σ coding coding Consider E 2 : S 1 → S 1 such that f ( x ) = 2 x mod 1

coding E 2 is a factor of σ Expanding maps are factors of σ semiconjugacy semiconjugacy semiconjugacy f : X → X and g : Y → Y maps h : Y → X is a semiconjugacy from g to f if f ◦ h = h ◦ g we also say that f is a factor of g

coding E 2 is a factor of σ Expanding maps are factors of σ semiconjugacy semiconjugacy semiconjugacy g → Y Y h ↓ ↓ h X → X f f is a factor of g

coding E 2 is a factor of σ Expanding maps are factors of σ E 2 is a factor of σ E 2 is a factor of σ E 2 is a factor of σ E 2 is a factor of σ on Σ + 2 that is, there exists a continuous surjective h such that σ Σ + Σ + → 2 2 ↓ ↓ h h S 1 S 1 → E 2

coding E 2 is a factor of σ Expanding maps are factors of σ E 2 is a factor of σ the semiconjugacy h Let us define h : Σ + 2 → S 1

coding E 2 is a factor of σ Expanding maps are factors of σ E 2 is a factor of σ proof definition of h define ∞ � E − n h ( x ) = 2 (∆ x n ) n = 0

coding E 2 is a factor of σ Expanding maps are factors of σ E 2 is a factor of σ proof h is well defined 2 (∆ x n ) consists of 2 n intervals of length E − n 1 2 n + 1 N � E − n 2 (∆ x n ) n = 0 1 is an interval of length 2 N + 1 h is a well-defined function

coding E 2 is a factor of σ Expanding maps are factors of σ E 2 is a factor of σ proof h is a semiconjugacy h is continuous (excercise) h is surjective (excercise) h ◦ σ = E 2 ◦ h

coding E 2 is a factor of σ Expanding maps are factors of σ introduction general expanding maps general expanding maps now let f : S 1 → S 1 be a general expanding map suppose deg( f ) = 2 ⇒ there is only one fixed point p ⇒ there is only one point q � = p such that f ( q ) = p call ∆ 0 = [ p , q ] and ∆ 1 = [ q , p ]

coding E 2 is a factor of σ Expanding maps are factors of σ theorem expanding maps are factors of σ theorem f : S 1 → S 1 expanding map deg( f ) = 2 ⇒ f is a factor of σ on Σ + 2 2 → S 1 such that f n ( h ( x )) ∈ ∆ x n for all n ≥ 0 ∃ h : Σ +

coding E 2 is a factor of σ Expanding maps are factors of σ proof proof definition of h following the previous theorem, let us define ∞ � f − n (∆ x n ) h ( x ) = n = 0

coding E 2 is a factor of σ Expanding maps are factors of σ proof proof h is well defined N � f − n (∆ x n ) � = ∅ n = 0 is an interval (induction) f n ( ξ ) , f n ( η ) ∈ ∆ x n for all n ⇒ ξ = η

coding E 2 is a factor of σ Expanding maps are factors of σ proof proof h is a semiconjugacy h is continuous h is surjective f ◦ h = h ◦ σ

coding E 2 is a factor of σ Expanding maps are factors of σ proof hints hints define N � f − n (∆ x n ) ∆ x 0 x 1 ... x N := n = 0

coding E 2 is a factor of σ Expanding maps are factors of σ proof hints hints prove by induction ∆ x 0 ... x N = [ a N , b N ] with f N + 1 ( a N ) = f N + 1 ( b N ) = p f N + 1 is injective in ( a N , b N )