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KoMaL A Problems
KoMaL A Problems 2018/2019
A. 735
A. 735
Part of
KoMaL A Problems 2018/2019
Problems
(1)
Piecewise linear function
Source: KöMaL A. 735
12/20/2018
For any function
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1]\to [0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
, let
P
n
(
f
)
P_n (f)
P
n
(
f
)
denote the number of fixed points of the function
f
(
f
(
…
f
⏟
n
(
x
)
…
)
\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )
n
f
(
f
(
…
f
(
x
)
…
)
, i.e., the number of points
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
satisfying
f
(
f
(
…
f
⏟
n
(
x
)
…
)
=
x
\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )=x
n
f
(
f
(
…
f
(
x
)
…
)
=
x
. Construct a piecewise linear, continuous, surjective function
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1] \to [0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
such that for a suitable
2
<
A
<
3
2<A<3
2
<
A
<
3
, the sequence
P
n
(
f
)
A
n
\frac{P_n(f)}{A^n}
A
n
P
n
(
f
)
converges.Based on the 8th problem of the Miklós Schweitzer competition, 2018
algebra
function
unsolved