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

coding classification proof proof h is well-defined, case 2 ∃ x � = y such that h f ( x ) = h f ( y ) = x ⇒ ∃ N such that f N ( x ) = p f , with f ( p f ) = p f

coding classification proof h is well-defined, case 2

coding classification proof h is well-defined, case 2 f N ( x ) = p f

coding classification proof h is well-defined, case 2 f N ( x ) = p f h f ( x ) = x

coding classification proof h is well-defined, case 2 f N ( x ) = p f h f ( x ) = x ⇒ σ N ( x ) = 0000000 . . . or σ N ( x ) = 1111111 . . . ⇐

coding classification proof h is well-defined, case 2 f N ( x ) = p f h f ( x ) = x ⇒ σ N ( x ) = 0000000 . . . or σ N ( x ) = 1111111 . . . ⇐ otherwise f n ( x ) = p ∈ ∆ 01 ∪ ∆ 10 for some n ≥ N

coding classification proof h is well-defined, case 2 f N ( x ) = p f h f ( x ) = x ⇒ σ N ( x ) = 0000000 . . . or σ N ( x ) = 1111111 . . . ⇐ otherwise f n ( x ) = p ∈ ∆ 01 ∪ ∆ 10 for some n ≥ N → contradiction

coding classification proof proof h is well defined, case 2

coding classification proof proof h is well defined, case 2 h f ( x ) = h f ( y ) = x with x � = y

coding classification proof proof h is well defined, case 2 h f ( x ) = h f ( y ) = x with x � = y N ≥ 0 the first such that f N ( x ) = p

coding classification proof proof h is well defined, case 2 h f ( x ) = h f ( y ) = x with x � = y N ≥ 0 the first such that f N ( x ) = p ⇒ x n = 0 and y n = 1 for all n ≥ N

coding classification proof proof h is well defined, case 2 h f ( x ) = h f ( y ) = x with x � = y N ≥ 0 the first such that f N ( x ) = p ⇒ x n = 0 and y n = 1 for all n ≥ N x N − 1 = 1 and y N − 1 = 0

coding classification proof proof h is well defined, case 2 h f ( x ) = h f ( y ) = x with x � = y N ≥ 0 the first such that f N ( x ) = p ⇒ x n = 0 and y n = 1 for all n ≥ N x N − 1 = 1 and y N − 1 = 0 x n = y n for all n ≤ N − 2

coding classification proof proof h is well defined, case 2

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y )

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y ) h g ( x ) ∈ ∆ g x 0 ... x N − 2 10000 ...

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y ) h g ( x ) ∈ ∆ g x 0 ... x N − 2 10000 ... h g ( y ) ∈ ∆ g x 0 ... x N − 2 011111 ...

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y ) h g ( x ) ∈ ∆ g x 0 ... x N − 2 10000 ... h g ( y ) ∈ ∆ g x 0 ... x N − 2 011111 ... ⇒ h g ( x ) , h g ( y ) ∈ ∆ x 1 ... x N − 2 = [ a N − 2 , b N − 2 ]

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y ) h g ( x ) ∈ ∆ g x 0 ... x N − 2 10000 ... h g ( y ) ∈ ∆ g x 0 ... x N − 2 011111 ... ⇒ h g ( x ) , h g ( y ) ∈ ∆ x 1 ... x N − 2 = [ a N − 2 , b N − 2 ] g N − 1 is injective in ( a N − 2 , b N − 2 )

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y ) h g ( x ) ∈ ∆ g x 0 ... x N − 2 10000 ... h g ( y ) ∈ ∆ g x 0 ... x N − 2 011111 ... ⇒ h g ( x ) , h g ( y ) ∈ ∆ x 1 ... x N − 2 = [ a N − 2 , b N − 2 ] g N − 1 is injective in ( a N − 2 , b N − 2 ) ∃ ! r ∈ ( a N − 2 , b N − 2 ) such that g N − 1 ( r ) = q g

coding classification proof proof h is well defined, case 2 let us show h g ( x ) = h g ( y ) h g ( x ) ∈ ∆ g x 0 ... x N − 2 10000 ... h g ( y ) ∈ ∆ g x 0 ... x N − 2 011111 ... ⇒ h g ( x ) , h g ( y ) ∈ ∆ x 1 ... x N − 2 = [ a N − 2 , b N − 2 ] g N − 1 is injective in ( a N − 2 , b N − 2 ) ∃ ! r ∈ ( a N − 2 , b N − 2 ) such that g N − 1 ( r ) = q g h g ( x ) ∈ [ r , b N − 2 ] and h g ( y ) ∈ [ a N − 2 , r ]

coding classification proof proof h is well defined, case 2

coding classification proof proof h is well defined, case 2 n ≥ N g − N (∆ 0 ) h g ( x ) ∈ [ r , b N − 2 ] ∩ �

coding classification proof proof h is well defined, case 2 n ≥ N g − N (∆ 0 ) = r h g ( x ) ∈ [ r , b N − 2 ] ∩ �

coding classification proof proof h is well defined, case 2 n ≥ N g − N (∆ 0 ) = r h g ( x ) ∈ [ r , b N − 2 ] ∩ � n ≥ N g − N (∆ 1 ) h g ( y ) ∈ [ a N / 2 , r ] ∩ �

coding classification proof proof h is well defined, case 2 n ≥ N g − N (∆ 0 ) = r h g ( x ) ∈ [ r , b N − 2 ] ∩ � n ≥ N g − N (∆ 1 ) = r h g ( y ) ∈ [ a N / 2 , r ] ∩ �

coding classification proof proof h is well defined, case 2 n ≥ N g − N (∆ 0 ) = r h g ( x ) ∈ [ r , b N − 2 ] ∩ � n ≥ N g − N (∆ 1 ) = r h g ( y ) ∈ [ a N / 2 , r ] ∩ � ⇒ h is well-defined.

coding classification proof proof h is continuous

coding classification proof proof h is continuous let x be such that f n ( x ) � = p f for all n ≥ 0

coding classification proof proof h is continuous let x be such that f n ( x ) � = p f for all n ≥ 0 1 take N > 0 such that d ( x , y ) < 3 N ⇒ d ( h g ( x ) , h g ( y )) < ε

coding classification proof proof h is continuous let x be such that f n ( x ) � = p f for all n ≥ 0 1 take N > 0 such that d ( x , y ) < 3 N ⇒ d ( h g ( x ) , h g ( y )) < ε x = � ∞ n = 0 f − n (∆ x n ) is in the interior of � N n = 0 f − n (∆ x n )

coding classification proof proof h is continuous let x be such that f n ( x ) � = p f for all n ≥ 0 1 take N > 0 such that d ( x , y ) < 3 N ⇒ d ( h g ( x ) , h g ( y )) < ε x = � ∞ n = 0 f − n (∆ x n ) is in the interior of � N n = 0 f − n (∆ x n ) ⇒ there is δ > 0 such that d ( x , y ) < δ ⇒ d ( h ( x ) , h ( y )) < ε

coding classification proof proof h is continuous

coding classification proof proof h is continuous let x be such that f K ( x ) = p f for some K > 0

coding classification proof proof h is continuous let x be such that f K ( x ) = p f for some K > 0 ⇒ h − 1 ( x ) = { x , y } such that f

coding classification proof proof h is continuous let x be such that f K ( x ) = p f for some K > 0 ⇒ h − 1 ( x ) = { x , y } such that f x = x 0 . . . x K − 2 011111 . . . and y = x 0 . . . x K − 2 100000 . . .

coding classification proof proof h is continuous let x be such that f K ( x ) = p f for some K > 0 ⇒ h − 1 ( x ) = { x , y } such that f x = x 0 . . . x K − 2 011111 . . . and y = x 0 . . . x K − 2 100000 . . . take ε > 0 and N > 0 and take y > x

coding classification proof proof h is continuous let x be such that f K ( x ) = p f for some K > 0 ⇒ h − 1 ( x ) = { x , y } such that f x = x 0 . . . x K − 2 011111 . . . and y = x 0 . . . x K − 2 100000 . . . take ε > 0 and N > 0 and take y > x if y ∈ ∆ x 0 ... x K − 2 1000 (with N subsymbols)

coding classification proof proof h is continuous let x be such that f K ( x ) = p f for some K > 0 ⇒ h − 1 ( x ) = { x , y } such that f x = x 0 . . . x K − 2 011111 . . . and y = x 0 . . . x K − 2 100000 . . . take ε > 0 and N > 0 and take y > x if y ∈ ∆ x 0 ... x K − 2 1000 (with N subsymbols) then d ( h ( x ) , h ( y )) < ε

coding classification proof proof h is continuous