Table Of ContentSpringerBriefs in Mathematics
Alexander M. Blokh · Oleksandr M. Sharkovsky
Sharkovsky Ordering
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Alexander M. Blokh Oleksandr M. Sharkovsky
Sharkovsky Ordering
Alexander M. Blokh Oleksandr M. Sharkovsky
Department of Mathematics Institute of Mathematics
University of Alabama at Birmingham National Academy of Sciences of Ukraine
Birmingham, AL, USA Kiev, Ukraine
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs in Mathematics
ISBN 978-3-030-99123-4 ISBN 978-3-030-99125-8 (eBook)
https://doi.org/10.1007/978-3-030-99125-8
Mathematics Subject Classification: 37E05, 37E15, 39A28, 37B40
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
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Preface
In 1964, a paper “Coexistence of cycles of a continuous mapping of the line into
itself”, by O. M. Sharkovsky, appeared in the Ukrainian Mathematical Journal, vol.
16 (1964), pp. 61–71. The following order was introduced among all positive integers:
3(cid:2)5(cid:2)7(cid:2) ...2 · 3(cid:2)2 · 5(cid:2)2 · 7(cid:2)(cid:2)˙22 (cid:2)2(cid:2)1
and the following theorem was proven: if m (cid:2) n then for any continuous mapping
f of the real line into itself the existence of a cycle of the map f of period n follows
from the existence of a cycle of the map f of period m.
This led to the inception of a new direction of research in the theory of dynamical
systems called combinatorial dynamics. A number of new papers appeared in which
the authors attempted to make this result more precise by taking into account the
mutual location of points on the line, or by considering various types of trajectories
on the plane other than cycles (e.g., homoclinic trajectories), or by considering certain
types of maps in higher dimension. In particular, in 1979, a paper “On Sharkovsky’s
cycle coexistence order”, by Peter Kloeden, appeared in Bull. Austral. Math. Soc.,
vol. 29 (1979), pp. 171–177; in this paper, perhaps, the expressions “Sharkovsky
ordering” and “Sharkovsky theorem” were first introduced, and it was proven that a
similar theorem and the same ordering hold for the so-called triangular maps that
can be defined in any dimension.
This book is devoted to the above-quoted surprising mathematical result. As its
scope is limited, we discuss only selected developments and direct attention of the
interested reader to the book by L. Alseda, J. Llibre, and M. Misiurewicz “Combi-
national Dynamics and Entropy in Dimension One”, 2nd edition, Advanced Series
in Nonlinear Dynamics, 5 (2000), a true encyclopedia of the named subject, for a
wider context and further references.
Most of the book was written in collaboration by the authors. Chapter 5 “Historical
Remarks” is written by O. M. Sharkovsky. The book uses two different versions of
the full name of O. M. Sharkovsky. Namely, his full name in Ukrainian is Oleksandr
v
vi Preface
Mykolayovych Sharkovsky (O. M. Sharkovsky), and its Russian transliteration is
Alexander Nikolaevich Sharkovsky (A. N. Sharkovsky).
We are grateful to M. Yu. Matvijchuk, who was involved in writing Sects. 1.1–
1.3 of Chap. 1 in 2013 before he moved to Canada.
To conclude, we would like to express our gratitude to the referees of the
manuscript for useful remarks.
Birmingham, AL, USA Alexander Blokh
Kyiv, Ukraine Oleksandr Sharkovsky
October 2020
Contents
1 Coexistence of Cycles for Continuous Interval Maps ................ 1
1.1 Introduction ................................................ 1
1.2 Proof of Forcing Sh-Theorem ................................. 3
1.2.1 Loops of Intervals Force Periodic Orbits .................. 4
1.2.2 The Beginning of the Sh-order .......................... 6
1.2.3 Three Implies Everything .............................. 6
1.2.4 Minimal Cycles Imply Sh-weaker Periods ................ 7
1.2.5 Orbits with Sh-strongest Periods Form Simplest Cycles ..... 9
1.3 Proof of Realization Sh-Theorem .............................. 12
1.4 Stability of the Sh-ordering ................................... 13
1.5 Visualization of the Sh-ordering ............................... 15
References ...................................................... 16
2 Combinatorial Dynamics on the Interval .......................... 19
2.1 Introduction ................................................ 19
2.2 Permutations: Refinement of Cycles’ Coexistence ................ 20
2.3 Rotation Theory ............................................. 27
2.4 Coexistence of Homoclinic Trajectories and Stratification
of the Space of Maps ......................................... 34
2.4.1 Homoclinic Trajectories, Horseshoes, and L-Schemes ...... 35
2.4.2 Coexistence (of Homoclinic Trajectories) and Its
Stability: Powers of Maps with L-Scheme
and Homoclinic Trajectories ............................ 40
References ...................................................... 48
3 Coexistence of Cycles for One-Dimensional Spaces ................. 51
3.1 Circle Maps ................................................ 51
3.2 Maps of the n-od ............................................ 57
3.3 Other Graph Maps ........................................... 61
3.3.1 Graph-Realizable Sets of Periods ........................ 62
3.3.2 Trees ................................................ 64
3.3.3 Graphs With Exactly One Loop ......................... 67
vii
viii Contents
3.3.4 Figure Eight Graph .................................... 68
References ...................................................... 68
4 Multidimensional Dynamical Systems ............................. 71
4.1 Triangular Maps ............................................. 72
4.2 Cyclically Permuting Maps ................................... 74
4.3 Multidimensional Perturbations of One-Dimensional Maps ........ 77
4.4 Infinitely-Dimensional Dynamical Systems, Generated
by One-Dimensional Maps .................................... 79
4.5 Final Remarks .............................................. 82
4.5.1 Multivalued Maps ..................................... 83
4.5.2 Nonlinear Difference Equations ......................... 83
References ...................................................... 83
5 Historical Remarks .............................................. 85
Appendix .......................................................... 93
Chapter 1
Coexistence of Cycles for Continuous
Interval Maps
1.1 Introduction
Given a map f : X → X, a point x ∈ X is said to be periodic if there exists an
integer n > 0 such that points x, f (x),..., f n−1(x) are pairwise distinct while
f n(x) = x; in that case, n is said to be the (minimal) period of x under f .Thes et
{x, f (x),..., f n−1(x)} is called the (periodic) orbit (of x). We will interchangeably
use equivalent terms periodic orbit and cycle. Our main interest in the present book
is studying the coexistence of periodic points of various periods of self-mappings of
a closed interval and of the real line R.
In fact, it is easy to see that the results concerning coexistence of periods for
self-mappings of R can be deduced from the results on coexistence of periods of
self-mappings of a closed interval. Indeed, if a map g : R → R has a periodic orbit
A of period n then we can construct the composition r ◦ g : A → A where A is the
smallest interval containing A and r is the natural retraction of the real line onto
A =[u, v] (thus, r is the identity map on [u, v], collapses (∞, u] to u, and collapses
[v, ∞) to v). One can see that periodic orbits of g are in fact periodic orbits of f ;
hence, the results on coexistence of periods that hold for f will hold for g.
Because of the above, from now on we consider continuous self-mappings of a
closed interval (say, [0, 1]) to itself. Denote by Per( f ) the set of all periodic points
of f , and by P( f ) the set of all periods of all periodic points of f .
Let us to introduce the Sharkovsky order:
3 ≻ 5 ≻ 7 ≻ ··· ≻2 · 3 ≻ 2 · 5 ≻ 2 · 7 ≻ ...2 2 · 3 ≻ 22 · 5 ≻ 22 · 7 ≻ ...2 2 ≻ 2 ≻ 1.
The Sharkovsky theorem consists of two parts. In the first one, it is proven that for
self-mappings of a closed interval the presence of a periodic orbit of a certain period
m implies the presence of periodic points of all periods that are Sharkovsky-weaker
than m. The second part shows that every possible initial segment of the Sharkovsky
order is realized on a continuous interval map as the set of periods of its periodic
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 1
A. M. Blokh and O. M. Sharkovsky, Sharkovsky Ordering,
SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-99125-8_1
