Editing Sharkovskii's theorem (section)

==Generalizations and related results==

Sharkovskii also proved the converse theorem: every [[upper set]] of the above order is the set of periods for some continuous function from an interval to itself. In fact all such sets of periods are achieved by the family of functions <math>T_h:[0,1]\to[0,1]</math>, <math>x\mapsto\min(h,1-2|x-1/2|)</math> for <math>h\in[0,1]</math>, except for the empty set of periods which is achieved by <math>T:\mathbb R\to\mathbb R</math>, <math>x\mapsto x+1</math>.<ref>{{cite book|last1=Alsedà|first1=L.|title=Combinatorial dynamics and entropy in dimension one|last2=Llibre|first2=J.|last3=Misiurewicz|first3=M.|date=2000|publisher=World Scientific Publishing Company|isbn=978-981-02-4053-0|author-link3=Michał Misiurewicz}}</ref><ref>{{cite journal |first1=K. |last1=Burns |first2=B. |last2=Hasselblatt |title=The Sharkovsky theorem: A natural direct proof |journal=[[American Mathematical Monthly]] |volume=118 |issue=3 |pages=229–244 |year=2011|doi=10.4169/amer.math.monthly.118.03.229 |citeseerx=10.1.1.216.784 |s2cid=15523008 }}</ref>

On the other hand, with additional information on the combinatorial structure of the interval map acting on the points in a periodic orbit, a period-n point may force period-3 (and hence all periods).  Namely, if the orbit type (the cyclic permutation generated by the map acting on the points in the periodic orbit) has a so-called stretching pair, then this implies the existence of a periodic point of period-3.  It can be shown (in an asymptotic sense) that almost all cyclic permutations admit at least one stretching pair, and hence almost all orbit types imply period-3.<ref>{{cite journal |first1=Erik |last1=Lundberg |title=Almost all orbit types imply period-3 |journal=[[Topology and Its Applications]] |volume=154 |pages=2741–2744
 |year=2007 |issue=14 |doi=10.1016/j.topol.2007.05.009 |doi-access=free }}</ref>

[[Tien-Yien Li]] and [[James A. Yorke]] showed in 1975 that not only does the existence of a period-3 cycle imply the existence of cycles of all periods, but in addition it implies the existence of an uncountable infinitude of points that never map to any cycle ([[chaos (mathematics)|chaotic points]])—a property known as [[period three implies chaos]].<ref>{{cite journal |first1=T. Y. |last1=Li |first2=J. A. |last2=Yorke |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |issue= 10|pages=985–992 |year=1975 |jstor=2318254 |doi=10.1080/00029890.1975.11994008 |bibcode=1975AmMM...82..985L }}</ref>

Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find a [[circle map]] with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit.  For example, [[Peter Kloeden]] showed that Sharkovskii's theorem holds for triangular mappings, i.e., mappings for which the component {{math|''f<sub>i</sub>''}} depends only on the first {{math|''i''}} components {{math|''x<sub>1</sub>,..., x<sub>i</sub>''}}.<ref>{{cite journal |first=P. E. |last=Kloeden |title=On Sharkovsky's cycle coexistence ordering |journal=Bull. Austral. Math. Soc. |volume=20 |year=1979 |issue= 2|pages=171–178 |doi=10.1017/S0004972700010819 |doi-access=free }}</ref>