Théorie ergodique et mesures in nies

(X, B)T
X µ σ X
(X, B, T, µ)µ
A T A µ
A µ
µ X
T µ
T
(X, B, T, µ)A
(T−nA)n∈N
(X, B, T, µ)
T

(X, B, T, µ)µ
(X, B, T, µ)
fL1(X, µ)
lim
n→+∞
1
n
n−1
X
k=0
f(Tkx) = ZX fdµ µ −p.p.
(Tnx)n∈N
µ
(f(Tnx))n∈Nf
σ
A µ(A)>0Tnx
A A
(X, B, T, µ)σ
fL1(X, µ)
lim
n→+∞
n−1
X
k=0
f(Tkx)=+∞µ−p.p.,
lim
n→+∞
1
n
n−1
X
k=0
f(Tkx)=0 µ−p.p.

(X, B, T, µ)
σ f g L1(X, µ)
ZX
gdµ > 0
lim
n→+∞Pn−1
k=0 f(Tkx)
Pn−1
k=0 g(Tkx)=RXfdµ
RXgdµµ−p.p.
g
µZX fdµ
L1(X, µ)
(an)n∈N
lim
n→+∞
1
an(x)
n−1
X
k=0
f(Tkx) = ZX fdµ µ
(an(x))n∈Nx
(X, B, T, µ)
σ(an)n∈N
lim
n→+∞an= +∞lim n→+∞
an
n= 0
lim
n→+∞
1
an
n−1
X
k=0
f(Tkx) = +∞µ−p.p. ∀f∈L1(X, µ), f ≥0,
lim inf
n→+∞
1
an
n−1
X
k=0
f(Tkx)=0 µ−p.p. ∀f∈L1(X, µ), f ≥0.

Z
(R,B(R), TMA,Leb) TMA :x7→ [x]−1+4{x}0≤ {x}<1/2
TMA :x7→ [x]−2+4{x}1/2≤ {x}<1 [x]
x{x}X
([0,1],B([0,1]), Tα, µα)α≥0Tα:
x7→ x(1 + 2αxα) 0 ≤x < 1/2Tα:x7→ 2x−1 1/2≤x < 1
α≥0Tα
µαα≥1
Z(Sn)n∈N0
P(S1= 0) = 2P(S1= 1) = 2P(S1=−1) = 1/2
(R,B(R), TMA,Leb)
lim
n→+∞
P1
√n#{0≤k < n :Sk= 0} ≤ x=2 √πZx
0
e−u2
2du.
g= 1[0,1] x[0,1]
n−1
X
k=0
g(Tk
MAx)
#{0≤k < n :Sk= 0}
fL1(R,Leb) 
lim
n→+∞
1
n1
2+
n−1
X
k=0
f(Tn
MAx) = 0 Leb ,
lim
n→+∞
1
n1
2−
n−1
X
k=0
f(Tn
MAx)=+∞.
n−1/2