Markov Graphs and Sharkovsky’s Theorem
Let
f
:
I
→
I
be a continuous mapping with
I
an interval in the real line.
Consider the following ordering of the positive integers.
3
5
7
2
⋅
3
2
⋅
5
2
⋅
7
2
n
⋅
3
2
n
⋅
5
2
n
⋅
7
2
n
+
1
⋅
3
2
n
+
1
⋅
5
2
n
2
n
−
1
8
4
2
1
This ordering is known as the
Sharkovsky ordering
.
In this ordering if the map
f
has a periodic orbit of period
k
and
k
m
, then
f
also has a periodic orbit of period
m
.
This is known as
Sharkovsky’s Theorem
.
One basis for proving this theorem is by means of
Markov graphs
.
Suppose that
f
:
I
→
I
is as above and that
I
j
{
}
j
=
1
p
is a collection of subintervals of
I
which are
pairwise disjoint except possibly for their endpoints.
The Markov graph for this
collection of intervals is a directed graph with the vertices being the collection of
intervals
I
j
{
}
j
=
1
p
and with an arrow
I
i
→
I
j
precisely when
f
(
I
i
)
⊃
I
j
.
Theorem.
Suppose that
I
1
→
I
2
→
→
I
k
{
}
is a cycle in this Markov graph, that is,
I
1
=
I
k
.
Then there is a point
x
∈
I
1
such that
f
k
(
x
)
=
x
and
f
i
(
x
)
∈
I
i
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 BLOCK
 Calculus, Integers, Period, Periodic points of complex quadratic mappings, Markov graph

Click to edit the document details