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Miklós Schweitzer
2021 Miklós Schweitzer
5
Iteration of f converges almost everywhere
Iteration of f converges almost everywhere
Source: 2021 Miklos Schweitzer, P5
November 2, 2021
real analysis
Problem Statement
Let
f
(
x
)
=
1
+
cos
(
2
π
x
)
2
f(x)=\frac{1+\cos(2 \pi x)}{2}
f
(
x
)
=
2
1
+
c
o
s
(
2
π
x
)
, for
x
∈
R
x \in \mathbb{R}
x
∈
R
, and
f
n
=
f
∘
⋯
∘
f
⏟
n
f^n=\underbrace{ f \circ \cdots \circ f}_{n}
f
n
=
n
f
∘
⋯
∘
f
. Is it true that for Lebesgue almost every
x
x
x
,
lim
n
→
∞
f
n
(
x
)
=
1
\lim_{n \to \infty} f^n(x)=1
lim
n
→
∞
f
n
(
x
)
=
1
?
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