Chapitre 7 Dynamique déterministe expansive

x∈[0,1[ x= 1 x= 0
S1
S1:= R/Z
f:S1→S1

f:R→RC∞f(x+k) =
f(x) 1 ∀x∈R,∀k∈Zx0∈S1
x1=f(x0), x2=f(x1), xn=f(xn−1) = . . . =fn(x0).
f
∀x∈R, f0(x)>1.
x0, x0
0=x0+δx δx 1
x1=f(x0)
x0
1=f(x0 0) = f(x0+δx)'f(x0) + δx.f0(x0) + Oδx2
|x0
1−x1|=f0(x0)
| {z }
>1
.|x0
0−x0|+Oδx2
ε∈R
f(x)=2x+ε
2πsin (2πx) 1
f(x+k) = f(x)+2k=f(x) 1 f:S1→
S1
f0(x) = 2 + εcos (2πx)
−1< ε < 1f0(x)>1
y∈S1x0=f−1(y), x1=
f−1(y)f:x→y=f(x)f−1:y→xj=
f−1
j(y)j= 0,1
f:S1→S1
C∞
f(x) = kx +g(x)
k≥2g(x+ 1) = g(x)
f0(x) = k+g0(x)>1,∀x

x0x1=f(x0)x2=f(x1)
y=f(x)=2x1f(x) = 2x+ε
2πsin (2πx) 1 ε'0.2

y
f:(S1→S1
x→2x1
x0= 0.24→x1=f(x0) = 0.48→x2=
f(x1) = 0.96→x3=f(x0)=0.92
x0= 0.01100111011 . . .
x1=f(x0) = 0.1100111011 . . .
x2=f(x1) = 0.100111011 . . .
x3=f(x2) = 0.00111011 . . .
x= 0.ε0ε1ε2. . .
ε0= 0,1x I0=0,1
2I1=1 2,1
x0x0
0=x0+δx δx = 2−(n+1)
εnn n xn=
fn(x0)=0.εn. . . x0
n=fn(x0 0)=0.ε0 n. . . ε0 n6=εnxnx0 n
Ij
δx x0
nxn
n n = 10 δx = 2−11 = 5.10−4
δx 2
0,1 0 + 1 = 10
10 + 1 = 11 11 + 1 = 100
1→10 →100 →. . .
0.011001 . . .

n xn=fn(x0) = x0
x0n x0
f
x0g=
f+δf ˜xn=gn(x0)n
xn=fn(x0)
{gn(x0
0)}x0,n
{fn(x0)}x0,n
{gn(x0
0)}nx0 0
{fn(x0)}nx0x0
0
fn(x0)gn(x0
0)
h:S1→S1x0
0=
h(x0)gn(x0
0) = h(fn(x0)) n