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Sharkovsky

# Sharkovsky - Markov Graphs and Sharkovskys Theorem Let f I...

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Markov Graphs and Sharkovsky’s Theorem Let f : I I be a continuous mapping with I an interval in the real line. Consider the following ordering of the positive integers. 3 5 7  2 3 2 5 2 7  2 n 3 2 n 5 2 n 7   2 n + 1 3 2 n + 1 5  2 n 2 n 1  8 4 2 1 This ordering is known as the Sharkovsky ordering . In this ordering if the map f has a periodic orbit of period k and k m , then f also has a periodic orbit of period m . This is known as Sharkovsky’s Theorem . One basis for proving this theorem is by means of Markov graphs . Suppose that f : I I is as above and that I j { } j = 1 p is a collection of subintervals of I which are pairwise disjoint except possibly for their endpoints. The Markov graph for this collection of intervals is a directed graph with the vertices being the collection of intervals I j { } j = 1 p and with an arrow I i I j precisely when f ( I i ) I j . Theorem. Suppose that I 1 I 2 I k { } is a cycle in this Markov graph, that is, I 1 = I k . Then there is a point x I 1 such that f k ( x ) = x and f i ( x ) I i .

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Sharkovsky - Markov Graphs and Sharkovskys Theorem Let f I...

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