Periods of periodic orbits for vertex maps on...
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![Page 1: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/1.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Periods of periodic orbits for vertex maps ongraphs
Chris Bernhardt
Fairfield University
Fairfield
CT 06824
July 2, 2012
![Page 2: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/2.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Outline
1 Introduction
2 an example
3 an example – with orientation
4 basic properties
5 two lemmas
6 vertex maps on graphs
7 basic properties
8 two lemmas – redux
9 Sharkovsky ordering
10 Final remarks
![Page 3: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/3.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Introduction
One of the basic starting points for one-dimensioncombinatorial dynamics is Sharkovsky’s Theorem.
Theorem
Let f : R ! R be continuous. If f has a periodic point of least
period v then f also has a periodic point of least period m for
any m / v, where
1 / 2 / 4 / . . . . . . 28 / 20 / 12 / . . . 14 / 10 / 6 . . . 7 / 5 / 3.
![Page 4: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/4.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example
✓
E1
E2
E3
1 2 3 4
![Page 5: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/5.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example
✓
E1
E2
E3
1 2 3 4
E
1
E
2
E
3
![Page 6: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/6.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example
✓
E1
E2
E3
1 2 3 4
M =
0
@0 0 11 0 10 1 1
1
A
![Page 7: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/7.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
a basic result
Theorem
Let M be the Markov matrix associated to a directed graph
that has vertices labeled E
1
, . . . ,En, then the ijth entry of M
k
gives the number of walks of length k from Ej to Ei .
Corollary
The trace of M
kgives the total number of closed walks of
length k.
![Page 8: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/8.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
a basic result
Theorem
Let M be the Markov matrix associated to a directed graph
that has vertices labeled E
1
, . . . ,En, then the ijth entry of M
k
gives the number of walks of length k from Ej to Ei .
Corollary
The trace of M
kgives the total number of closed walks of
length k.
![Page 9: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/9.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example – with orientation
✓
>E1
>E2
>E3
1 2 3 4
+
+�
�
�
![Page 10: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/10.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example – with orientation
✓
>E1
>E2
>E3
1 2 3 4
E
1
E
2
E
3
+
+�
�
�
![Page 11: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/11.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example – with orientation
✓
>E1
>E2
>E3
1 2 3 4
M
0
(✓) =
0
BB@
0 1 0 00 0 1 00 0 0 11 0 0 0
1
CCA ,M1
(✓) =
0
@0 0 �11 0 �10 1 �1
1
A
![Page 12: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/12.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
an example – with orientation
✓
>E1
>E2
>E3
1 2 3 4
M
0
(✓) =
0
BB@
0 0 0 11 0 0 00 1 0 00 0 1 0
1
CCA ,M1
(✓) =
0
@0 0 �11 0 �10 1 �1
1
A
![Page 13: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/13.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
The ijth entry of (M1
(✓))k gives the number of positively
oriented walks of length k from Ej to Ei minus the number
negatively oriented walks from Ej to Ei .
Corollary
The trace of (M1
(✓))k gives the number of positively oriented
closed walks of length k minus the number of negatively
oriented closed walks of length k.
![Page 14: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/14.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
The ijth entry of (M1
(✓))k gives the number of positively
oriented walks of length k from Ej to Ei minus the number
negatively oriented walks from Ej to Ei .
Corollary
The trace of (M1
(✓))k gives the number of positively oriented
closed walks of length k minus the number of negatively
oriented closed walks of length k.
![Page 15: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/15.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
1 (M0
(✓))k = M
0
(✓k)
2 (M1
(✓))k = M
1
(✓k)
✓
>E1
>E2
>E3
1 2 3 4
M
0
(✓) =
0
BB@
0 0 0 11 0 0 00 1 0 00 0 1 0
1
CCA ,M1
(✓) =
0
@0 0 �11 0 �10 1 �1
1
A
![Page 16: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/16.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
1 (M0
(✓))k = M
0
(✓k)
2 (M1
(✓))k = M
1
(✓k)
✓
>E1
>E2
>E3
1 2 3 4
M
0
(✓) =
0
BB@
0 0 0 11 0 0 00 1 0 00 0 1 0
1
CCA ,M1
(✓) =
0
@0 0 �11 0 �10 1 �1
1
A
![Page 17: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/17.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
Trace (M0
(✓))�Trace (M1
(✓)) = 1.
![Page 18: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/18.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma
Lemma
Let f : R ! R be continuous. Suppose that f has a periodic
point of period 17. Then f has a periodic point of period 2k
for any non-negative integer k.
![Page 19: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/19.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma
Proof.
Since 17 is not a divisor of 2k we know that ✓2kdoes not fix
any of the integers in {1, 2, . . . , 17}. So Trace (M0
(✓2k)) = 0.
So Trace (M1
(✓2k)) = �1. So the oriented Markov graph has a
vertex Ej with a closed walk from Ej to itself of length 2k withnegative orientation. Since the orientation is negative it cannotbe the repetition of a shorter closed walk, as any shorter closedwalk would have to be repeated an even number of times. Sothere is a periodic point in Ej with minimum period 2k .
![Page 20: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/20.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma
Proof.
Since 17 is not a divisor of 2k we know that ✓2kdoes not fix
any of the integers in {1, 2, . . . , 17}. So Trace (M0
(✓2k)) = 0.
So Trace (M1
(✓2k)) = �1.
So the oriented Markov graph has avertex Ej with a closed walk from Ej to itself of length 2k withnegative orientation. Since the orientation is negative it cannotbe the repetition of a shorter closed walk, as any shorter closedwalk would have to be repeated an even number of times. Sothere is a periodic point in Ej with minimum period 2k .
![Page 21: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/21.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma
Proof.
Since 17 is not a divisor of 2k we know that ✓2kdoes not fix
any of the integers in {1, 2, . . . , 17}. So Trace (M0
(✓2k)) = 0.
So Trace (M1
(✓2k)) = �1. So the oriented Markov graph has a
vertex Ej with a closed walk from Ej to itself of length 2k withnegative orientation.
Since the orientation is negative it cannotbe the repetition of a shorter closed walk, as any shorter closedwalk would have to be repeated an even number of times. Sothere is a periodic point in Ej with minimum period 2k .
![Page 22: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/22.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma
Proof.
Since 17 is not a divisor of 2k we know that ✓2kdoes not fix
any of the integers in {1, 2, . . . , 17}. So Trace (M0
(✓2k)) = 0.
So Trace (M1
(✓2k)) = �1. So the oriented Markov graph has a
vertex Ej with a closed walk from Ej to itself of length 2k withnegative orientation. Since the orientation is negative it cannotbe the repetition of a shorter closed walk, as any shorter closedwalk would have to be repeated an even number of times.
Sothere is a periodic point in Ej with minimum period 2k .
![Page 23: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/23.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma
Proof.
Since 17 is not a divisor of 2k we know that ✓2kdoes not fix
any of the integers in {1, 2, . . . , 17}. So Trace (M0
(✓2k)) = 0.
So Trace (M1
(✓2k)) = �1. So the oriented Markov graph has a
vertex Ej with a closed walk from Ej to itself of length 2k withnegative orientation. Since the orientation is negative it cannotbe the repetition of a shorter closed walk, as any shorter closedwalk would have to be repeated an even number of times. Sothere is a periodic point in Ej with minimum period 2k .
![Page 24: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/24.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma
Lemma
Let f : R ! R be continuous. Suppose that f has a periodic
point of period 17. Then f has a periodic point of period m for
any non-negative integer m > 17.
![Page 25: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/25.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma
Proof.
Trace (M1
(✓)) = �1. So there vertex Ej in the Markov graphwith a closed walk of length one with negative orientation.
M
1
(✓)17 is the identity matrix. So there is a closed walk fromEj to itself with length 17 and with positive orientation. Theclosed walk of length 17 is not a repetition of the walk oflength 1. We can construct a non-repetitive closed walk oflength m by going once around the walk of length 17 and thenm � 17 times around the walk of length 1.
![Page 26: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/26.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma
Proof.
Trace (M1
(✓)) = �1. So there vertex Ej in the Markov graphwith a closed walk of length one with negative orientation.M
1
(✓)17 is the identity matrix. So there is a closed walk fromEj to itself with length 17 and with positive orientation.
Theclosed walk of length 17 is not a repetition of the walk oflength 1. We can construct a non-repetitive closed walk oflength m by going once around the walk of length 17 and thenm � 17 times around the walk of length 1.
![Page 27: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/27.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma
Proof.
Trace (M1
(✓)) = �1. So there vertex Ej in the Markov graphwith a closed walk of length one with negative orientation.M
1
(✓)17 is the identity matrix. So there is a closed walk fromEj to itself with length 17 and with positive orientation. Theclosed walk of length 17 is not a repetition of the walk oflength 1.
We can construct a non-repetitive closed walk oflength m by going once around the walk of length 17 and thenm � 17 times around the walk of length 1.
![Page 28: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/28.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma
Proof.
Trace (M1
(✓)) = �1. So there vertex Ej in the Markov graphwith a closed walk of length one with negative orientation.M
1
(✓)17 is the identity matrix. So there is a closed walk fromEj to itself with length 17 and with positive orientation. Theclosed walk of length 17 is not a repetition of the walk oflength 1. We can construct a non-repetitive closed walk oflength m by going once around the walk of length 17 and thenm � 17 times around the walk of length 1.
![Page 29: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/29.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
vertex maps on graphs
v
1
v
2
v
3
v
4
v
5
E
1
E
2
E
3
E
4
E
5
E
6
M
0
(✓) =
0
BBBB@
0 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 0
1
CCCCA
![Page 30: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/30.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
vertex maps on graphs
v
1
v
2
v
3
v
4
v
5
E
1
E
2
E
3
E
4
E
5
E
6
M
0
(✓) =
0
BBBB@
0 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 0
1
CCCCA
![Page 31: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/31.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
vertex maps on graphs
v
1
v
2
v
3
v
4
v
5
E
1
E
2
E
3
E
4
E
5
E
6
M
1
(✓) =
0
BBBBBB@
0 0 0 0 0 00 �1 0 1 0 �11 �1 0 0 1 00 0 0 0 �1 00 0 �1 0 �1 �10 1 0 �1 0 1
1
CCCCCCA
![Page 32: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/32.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
1 (M0
(✓))k = M
0
(✓k)
2 (M1
(✓))k = M
1
(✓k)
3
Trace (M0
(✓))�Trace (M1
(✓)) = Lf
Corollary
If the underlying map is homotopic to the identity, then
Trace (M0
(✓))�Trace (M1
(✓)) = v � e
![Page 33: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/33.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
1 (M0
(✓))k = M
0
(✓k)
2 (M1
(✓))k = M
1
(✓k)
3
Trace (M0
(✓))�Trace (M1
(✓)) = Lf
Corollary
If the underlying map is homotopic to the identity, then
Trace (M0
(✓))�Trace (M1
(✓)) = v � e
![Page 34: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/34.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
basic properties
Theorem
1 (M0
(✓))k = M
0
(✓k)
2 (M1
(✓))k = M
1
(✓k)
3
Trace (M0
(✓))�Trace (M1
(✓)) = Lf
Corollary
If the underlying map is homotopic to the identity, then
Trace (M0
(✓))�Trace (M1
(✓)) = v � e
![Page 35: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/35.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Lemma
Let G be a graph and f a vertex map from G to itself that is
homotopic to the identity. Suppose that the vertices form one
periodic orbit. Suppose f flips an edge. If v is not a divisor of
2k , then f has a periodic point with period 2k .
![Page 36: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/36.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Proof.
Since f flips an edge, there must be at least one loop in theMarkov graph that has length 1 and has negative orientation.
Since Trace(M1
(f )) = e � v , there must be at least e � v + 1loops in Markov graph of length 1 that have positiveorientation. By going around each of these loops in the Markovgraph twice we can see that there must be at least e � v + 2loops of length 2 that have positive orientation.Since Trace(M
1
(f )2) = e � v , there must be at least one loopof length 2 with negative orientation. Since it has negativeorientation, it cannot be the repetition of a shorter loop. Sothe Markov graph of f has a non-repetitive loop of length 2with negative orientation.etc – use induction
![Page 37: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/37.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Proof.
Since f flips an edge, there must be at least one loop in theMarkov graph that has length 1 and has negative orientation.Since Trace(M
1
(f )) = e � v , there must be at least e � v + 1loops in Markov graph of length 1 that have positiveorientation.
By going around each of these loops in the Markovgraph twice we can see that there must be at least e � v + 2loops of length 2 that have positive orientation.Since Trace(M
1
(f )2) = e � v , there must be at least one loopof length 2 with negative orientation. Since it has negativeorientation, it cannot be the repetition of a shorter loop. Sothe Markov graph of f has a non-repetitive loop of length 2with negative orientation.etc – use induction
![Page 38: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/38.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Proof.
Since f flips an edge, there must be at least one loop in theMarkov graph that has length 1 and has negative orientation.Since Trace(M
1
(f )) = e � v , there must be at least e � v + 1loops in Markov graph of length 1 that have positiveorientation. By going around each of these loops in the Markovgraph twice we can see that there must be at least e � v + 2loops of length 2 that have positive orientation.
Since Trace(M1
(f )2) = e � v , there must be at least one loopof length 2 with negative orientation. Since it has negativeorientation, it cannot be the repetition of a shorter loop. Sothe Markov graph of f has a non-repetitive loop of length 2with negative orientation.etc – use induction
![Page 39: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/39.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Proof.
Since f flips an edge, there must be at least one loop in theMarkov graph that has length 1 and has negative orientation.Since Trace(M
1
(f )) = e � v , there must be at least e � v + 1loops in Markov graph of length 1 that have positiveorientation. By going around each of these loops in the Markovgraph twice we can see that there must be at least e � v + 2loops of length 2 that have positive orientation.Since Trace(M
1
(f )2) = e � v , there must be at least one loopof length 2 with negative orientation. Since it has negativeorientation, it cannot be the repetition of a shorter loop.
Sothe Markov graph of f has a non-repetitive loop of length 2with negative orientation.etc – use induction
![Page 40: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/40.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Proof.
Since f flips an edge, there must be at least one loop in theMarkov graph that has length 1 and has negative orientation.Since Trace(M
1
(f )) = e � v , there must be at least e � v + 1loops in Markov graph of length 1 that have positiveorientation. By going around each of these loops in the Markovgraph twice we can see that there must be at least e � v + 2loops of length 2 that have positive orientation.Since Trace(M
1
(f )2) = e � v , there must be at least one loopof length 2 with negative orientation. Since it has negativeorientation, it cannot be the repetition of a shorter loop. Sothe Markov graph of f has a non-repetitive loop of length 2with negative orientation.
etc – use induction
![Page 41: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/41.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
first lemma – redux
Proof.
Since f flips an edge, there must be at least one loop in theMarkov graph that has length 1 and has negative orientation.Since Trace(M
1
(f )) = e � v , there must be at least e � v + 1loops in Markov graph of length 1 that have positiveorientation. By going around each of these loops in the Markovgraph twice we can see that there must be at least e � v + 2loops of length 2 that have positive orientation.Since Trace(M
1
(f )2) = e � v , there must be at least one loopof length 2 with negative orientation. Since it has negativeorientation, it cannot be the repetition of a shorter loop. Sothe Markov graph of f has a non-repetitive loop of length 2with negative orientation.etc – use induction
![Page 42: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/42.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma – redux
Lemma
Let G be a graph and f a map from G to itself that is
homotopic to the identity. Suppose that the vertices form one
periodic orbit. Suppose f flips an edge. If v = 2pq, whereq > 1 is odd and p � 0, then f has a periodic point with
period 2pr for any r � q.
Proof.
Similar trace argument.
![Page 43: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/43.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
second lemma – redux
Lemma
Let G be a graph and f a map from G to itself that is
homotopic to the identity. Suppose that the vertices form one
periodic orbit. Suppose f flips an edge. If v = 2pq, whereq > 1 is odd and p � 0, then f has a periodic point with
period 2pr for any r � q.
Proof.
Similar trace argument.
![Page 44: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/44.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Sharkovsky ordering
The Sharkovsky ordering can be defined as follows:(what positive integers does v force?)
1 2l / 2k = v if l k .
2 If v = 2ks, where s > 1 is odd, then1 2l / v , for all positive integers l .2 2k r / v , where r � s.3 2l r / v , where l > k and such that 2l r < v .
![Page 45: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/45.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Sharkovsky ordering
The Sharkovsky ordering can be defined as follows:(what positive integers does v force?)
1 2l / 2k = v if l k .2 If v = 2ks, where s > 1 is odd, then
1 2l / v , for all positive integers l .2 2k r / v , where r � s.3 2l r / v , where l > k and such that 2l r < v .
![Page 46: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/46.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Sharkovsky ordering
The Sharkovsky ordering can be defined as follows:(what positive integers does v force?)
1 2l / 2k = v if l k .2 If v = 2ks, where s > 1 is odd, then
1 2l / v , for all positive integers l .
2 2k r / v , where r � s.3 2l r / v , where l > k and such that 2l r < v .
![Page 47: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/47.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Sharkovsky ordering
The Sharkovsky ordering can be defined as follows:(what positive integers does v force?)
1 2l / 2k = v if l k .2 If v = 2ks, where s > 1 is odd, then
1 2l / v , for all positive integers l .2 2k r / v , where r � s.
3 2l r / v , where l > k and such that 2l r < v .
![Page 48: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/48.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
Sharkovsky ordering
The Sharkovsky ordering can be defined as follows:(what positive integers does v force?)
1 2l / 2k = v if l k .2 If v = 2ks, where s > 1 is odd, then
1 2l / v , for all positive integers l .2 2k r / v , where r � s.3 2l r / v , where l > k and such that 2l r < v .
![Page 49: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/49.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
final remarks
1 This is a way of generalizing from maps of the interval andcircle to maps on graphs.
2 This is not the most general method of generalizing, but itleads to interesting results, and is very accessible.
3 More info at: Sharkovsky’s theorem and one-dimensional
combinatorial dynamics arxiv.org/abs/1201.3583
Thank you!
![Page 50: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/50.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
final remarks
1 This is a way of generalizing from maps of the interval andcircle to maps on graphs.
2 This is not the most general method of generalizing, but itleads to interesting results, and is very accessible.
3 More info at: Sharkovsky’s theorem and one-dimensional
combinatorial dynamics arxiv.org/abs/1201.3583
Thank you!
![Page 51: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/51.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
final remarks
1 This is a way of generalizing from maps of the interval andcircle to maps on graphs.
2 This is not the most general method of generalizing, but itleads to interesting results, and is very accessible.
3 More info at: Sharkovsky’s theorem and one-dimensional
combinatorial dynamics arxiv.org/abs/1201.3583
Thank you!
![Page 52: Periods of periodic orbits for vertex maps on graphsmat.uab.cat/~alseda/AIMS2012-SS23/Bernhardt.pdfPeriods of periodic orbits for vertex maps on graphs Introduction an example an example](https://reader033.fdocuments.us/reader033/viewer/2022060922/60ae0f589525d23ed35b5a56/html5/thumbnails/52.jpg)
Periods of
periodic orbits
for vertex
maps on
graphs
Introduction
an example
an example –
with
orientation
basic
properties
two lemmas
vertex maps
on graphs
basic
properties
two lemmas –
redux
Sharkovsky
ordering
Final remarks
final remarks
1 This is a way of generalizing from maps of the interval andcircle to maps on graphs.
2 This is not the most general method of generalizing, but itleads to interesting results, and is very accessible.
3 More info at: Sharkovsky’s theorem and one-dimensional
combinatorial dynamics arxiv.org/abs/1201.3583
Thank you!