The Sharkovsky Theorem: A Natural Direct Proof

  title={The Sharkovsky Theorem: A Natural Direct Proof},
  author={Keith Burns and Boris Hasselblatt},
  journal={The American Mathematical Monthly},
  pages={229 - 244}
Abstract We give a natural and direct proof of a famous result by Sharkovsky that gives a complete description of possible sets of periods for interval maps. The new ingredient is the use of Štefan sequences. 
A Sharkovsky theorem for non-locally connected spaces
We extend Sharkovsky's Theorem to several new classes of spaces, which include some well-known examples of non-locally connected continua, such as the topologist's sine curve and the Warsaw circle.
Periodic Points of Some Discontinuous Mappings
For some discontinuous mappings, an analogue of the Sharkovsky theorem is proved, which proves that periodic points of discontinuity mappings have periodic points.
An Interesting Application of the Intermediate Value Theorem: A Simple Proof of Sharkovsky's Theorem
This note is intended primarily for college calculus students right after the introduction of the Intermediate Value Theorem, to show them how the Intermediate Value Theorem is used repeatedly and
An Introductory Look at Deterministic Chaos
Samuel Coskey Mathematics Bachelor of Science An Introductory Look at Deterministic Chaos by Kenneth Coiteux This is a brief introduction to deterministic chaos. We will be studying the logistic map
From the Sharkovskii theorem to periodic orbits for the Rössler system
Symbolic dynamics for non-uniformly hyperbolic systems
  • Yuri Lima
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2020
Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory
Symbolic dynamics for nonuniformly hyperbolic systems
This survey describes the recent advances in the construction of Markov partitions for nonuniformly hyperbolic systems. One important feature of this development comes from a finer theory of
About Chaotic Dynamics in the Twisted Horseshoe Map
A simple and rigorous proof based on a different approach is given that this twisted horseshoe map presents chaotic dynamics and the possibility of getting chaotic dynamics for a broader class of maps is highlighted.
One-dimensional dynamical systems
The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally


A collection of simple proofs of Sharkovsky's theorem
Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.
A Simple Special Case of Sharkovskii's Theorem
In this note I c R is a bounded closed interval and f: I -> I is a continuous map; fn denotes the n-fold composition of f with itself. A point x E I is a periodic point for f with period p if fP(x) =
A Simple Proof of Sharkovsky's Theorem Revisited
Sharkovsky's theorem states that, if / is a continuous map from a compact interval into itself that has a period-m point, then / also has aperiod-^ point whenever m < n in the Sharkovsky's ordering of the natural numbers.
The basic result of this investigation may be formulated as follows. Consider the set of natural numbers in which the following relationship is introduced: n1 precedes n2 (n1 ≼ n2) if for any
A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line
Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT
A Simple Proof of Sharkovsky's Theorem
1. M. Berger, Geometry, vol. 2, Springer-Verlag, New York, 1987. 2. , Encounter with a geometer II, Notices Amer. Math. Soc. 47 (2000) 326–340. 3. J. Cheeger and D. Ebin, Comparison Theorems in
Combinatorial Dynamics and Entropy in Dimension One
Preliminaries: general notation graphs, loops and cycles. Interval maps: the Sharkovskii Theorem maps with the prescribed set of periods forcing relation patterns for interval maps antisymmetry of
Periodic Points of Continuous Functions
Analysis of the cyclic behavior of points under repeated application of a function yields insights into population patterns.
Topological Dynamics on the Interval
A great deal has been written about maps of the interval, especially in recent years. In addition to many detailed mathematical papers, there have been a number of numerical studies and descriptive