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

coding the shift transformation example example . . . x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 1 0 1 1 0 0 1 0 0 0 1 1 . . . σ ( x ) 0 1 1 0 0 1 0 0 0 1 1 0 . . . σ 2 ( x ) 1 1 0 0 1 0 0 0 1 1 0 1 . . . σ 3 ( x ) 1 0 0 1 0 0 0 1 1 0 1 1 . . . σ 4 ( x ) . . . 0 0 1 0 0 0 1 1 0 1 1 0 σ 5 ( x ) 0 1 0 0 0 1 1 0 1 1 0 1 . . .

coding the shift transformation example example x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 . . . x 1 0 1 1 0 0 1 0 0 0 1 1 . . . σ ( x ) 0 1 1 0 0 1 0 0 0 1 1 0 . . . σ 2 ( x ) 1 1 0 0 1 0 0 0 1 1 0 1 . . . σ 3 ( x ) 1 0 0 1 0 0 0 1 1 0 1 1 . . . σ 4 ( x ) 0 0 1 0 0 0 1 1 0 1 1 0 . . . σ 5 ( x ) 0 1 0 0 0 1 1 0 1 1 0 1 . . . . . . . . .

coding the shift transformation properties of the shift Index coding 1 coding the space Σ + 2 the shift transformation 2 properties of the shift

coding the shift transformation properties of the shift fixed points fixed point x is a fixed point if σ ( x ) = x

coding the shift transformation properties of the shift fixed points x is a fixed point

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases:

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 00

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 000

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000 . . .

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000 . . . 1

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000 . . . 11

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000 . . . 111

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000 . . . 1111

coding the shift transformation properties of the shift fixed points x is a fixed point ⇒ σ ( x ) = x ⇒ [ σ ( x )] n = x n for all n ≥ 0 ⇒ x n + 1 = x n for all n ≥ 0 two cases: x 0 = 0 1 x 0 = 1 2 0000 . . . 1111 . . .

coding the shift transformation properties of the shift periodic points periodic point x is a periodic point if ∃ N ≥ 0 such that x , σ ( x ) , σ 2 ( x ) , . . . , σ N ( x ) = x o ( x ) :

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4 (2) 01

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4 (2) 010

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4 (2) 0101

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4 (2) 01010

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4 (2) 010101

coding the shift transformation properties of the shift periodic points of period 2 x is a periodic point of period 2 ⇒ σ 2 ( x ) = x ⇐ ⇒ [ σ 2 ( x )] n = x n for each n ≥ 0 ⇐ ⇐ ⇒ x n + 2 = x n for all n ≥ 0 4 cases x 0 x 1 = 00 1 x 0 x 1 = 01 2 x 0 x 1 = 10 3 x 0 x 1 = 11 4 (2) 010101 . . .

coding the shift transformation properties of the shift periodic point are dense periodic points are dense the periodic points for the shift transformation are dense in Σ + 2

coding the shift transformation properties of the shift transitivity transitivity the shift transformation is transitive

coding the shift transformation properties of the shift hint

coding the shift transformation properties of the shift hint there is x with dense orbit:

coding the shift transformation properties of the shift hint there is x with dense orbit: x =

coding the shift transformation properties of the shift hint there is x with dense orbit: x = 0

coding the shift transformation properties of the shift hint there is x with dense orbit: x = 0 1

coding the shift transformation properties of the shift hint there is x with dense orbit: x = 0 1 00

coding the shift transformation properties of the shift hint there is x with dense orbit: x = 0 1 00 01

coding the shift transformation properties of the shift hint there is x with dense orbit: x = 0 1 00 01 10

coding the shift transformation properties of the shift hint there is x with dense orbit: x = 0 1 00 01 10 11