The Sharkovsky Theorem: A Natural Direct Proof

@article{Burns2011TheST,
  title={The Sharkovsky Theorem: A Natural Direct Proof},
  author={Keith Burns and Boris Hasselblatt},
  journal={The American Mathematical Monthly},
  year={2011},
  volume={118},
  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. 
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References

SHOWING 1-10 OF 38 REFERENCES
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
TLDR
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.
COEXISTENCE OF CYCLES OF A CONTINUOUS MAP OF THE LINE INTO ITSELF
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
...
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