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

coding the shift transformation

Dynamical systems

Expanding maps on the circle. Coding Jana Rodriguez Hertz

ICTP

2018

coding the shift transformation coding

Index

1

coding coding the space Σ+

2 2

the shift transformation properties of the shift

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x =

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 000

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 000

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0000

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0000

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00001

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00001

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 000011

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 000011

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0000110

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0000110

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00001100

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00001100

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 000011001

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 000011001

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0000110011

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 0000110011

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00001100110011

coding the shift transformation coding

coding

Consider f : S1 → S1 such that f(x) = 2x mod 1 x = 00001100110011 . . .

coding the shift transformation the space Σ+

2

Index

1

coding coding the space Σ+

2 2

the shift transformation properties of the shift

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2 = {0, 1}N

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2 = {0, 1}N

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2 = {0, 1}N

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2 = {0, 1}N

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2 = {0, 1}N

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

the space Σ+

2

Σ+

2 = {0, 1}N

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 . . . . . . . . .

coding the shift transformation the space Σ+

2

a metric on Σ+

2

We can define a metric on Σ+

2 :

coding the shift transformation the space Σ+

2

a metric on Σ+

2

We can define a metric on Σ+

2 :

d(x, y) =

coding the shift transformation the space Σ+

2

a metric on Σ+

2

We can define a metric on Σ+

2 :

d(x, y) =

∞

• n=0

|xn − yn| 3n+1

coding the shift transformation the space Σ+

2

a metric on Σ+

2

We can define a metric on Σ+

2 :

d(x, y) =

∞

• n=0

|xn − yn| 3n+1

Proposition

(Σ+

2 , d) is a compact metric space

coding the shift transformation the space Σ+

2

a metric on Σ+

2

We can define a metric on Σ+

2 :

d(x, y) =

∞

• n=0

|xn − yn| 3n+1

Proposition

(Σ+

2 , d) is a compact metric space

d(x, y) < 1/

3n+1

⇔ xi = yi for i = 0, . . . , n

coding the shift transformation the space Σ+

2

example

example

points in B(1, 1/

36)

coding the shift transformation the space Σ+

2

example

example

points in B(1, 1/

36)

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

example

example

points in B(1, 1/

36)

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

example

example

points in B(1, 1/

36)

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

example

example

points in B(1, 1/

36)

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1 1 . . .

coding the shift transformation the space Σ+

2

example

example

points in B(1, 1/

36)

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1 1 . . . . . . . . .

coding the shift transformation

the shift transformation

the shift transformation

the shift transformation σ : Σ+ → Σ+

coding the shift transformation

the shift transformation

the shift transformation

the shift transformation σ : Σ+ → Σ+ is defined by

coding the shift transformation

the shift transformation

the shift transformation

the shift transformation σ : Σ+ → Σ+ is defined by [σ(x)]n = xn+1

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . .

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . . σ(x) 1 1 1 1 1 . . .

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . . σ(x) 1 1 1 1 1 . . . σ2(x) 1 1 1 1 1 1 . . .

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . . σ(x) 1 1 1 1 1 . . . σ2(x) 1 1 1 1 1 1 . . . σ3(x) 1 1 1 1 1 1 . . .

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . . σ(x) 1 1 1 1 1 . . . σ2(x) 1 1 1 1 1 1 . . . σ3(x) 1 1 1 1 1 1 . . . σ4(x) 1 1 1 1 1 . . .

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . . σ(x) 1 1 1 1 1 . . . σ2(x) 1 1 1 1 1 1 . . . σ3(x) 1 1 1 1 1 1 . . . σ4(x) 1 1 1 1 1 . . . σ5(x) 1 1 1 1 1 1 . . .

coding the shift transformation

example

example

x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 . . . x 1 1 1 1 1 1 . . . σ(x) 1 1 1 1 1 . . . σ2(x) 1 1 1 1 1 1 . . . σ3(x) 1 1 1 1 1 1 . . . σ4(x) 1 1 1 1 1 . . . σ5(x) 1 1 1 1 1 1 . . . . . . . . .

coding the shift transformation properties of the shift

Index

1

coding coding the space Σ+

2 2

the shift transformation 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 = xn 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 = xn for all n ≥ 0 ⇒ xn+1 = xn 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 = xn for all n ≥ 0 ⇒ xn+1 = xn 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 = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

00

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

000

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

0000

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

0000 . . .

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

0000 . . . 1

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

0000 . . . 11

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

0000 . . . 111

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

0000 . . . 1111

coding the shift transformation properties of the shift

fixed points

x is a fixed point ⇒ σ(x) = x ⇒ [σ(x)]n = xn for all n ≥ 0 ⇒ xn+1 = xn for all n ≥ 0 two cases:

1

x0 = 0

2

x0 = 1

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) :

x, σ(x), σ2(x), . . . , σN(x) = 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 ⇐ ⇒ σ2(x) = x

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn 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 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn 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 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn 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 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

(2) 01

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

(2) 010

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

(2) 0101

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

(2) 01010

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

(2) 010101

coding the shift transformation properties of the shift

periodic points of period 2

x is a periodic point of period 2 ⇐ ⇒ σ2(x) = x ⇐ ⇒ [σ2(x)]n = xn for each n ≥ 0 ⇐ ⇒ xn+2 = xn for all n ≥ 0 4 cases

1

x0x1 = 00

2

x0x1 = 01

3

x0x1 = 10

4

x0x1 = 11

(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

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000 001

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000 001 010

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000 001 010 011

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000 001 010 011 100

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000 001 010 011 100 . . .

coding the shift transformation properties of the shift

hint

there is x with dense orbit: x = 0 1 00 01 10 11 000 001 010 011 100 . . . x contains all finite sequences of 0’s and 1’s