Sharkovsky ordering

More simple proofs of Sharkovsky ’ s theorem

The Sharkovsky Theorem: A Natural Direct Proof

We give a proof of the Sharkovsky Theorem that is selfcontained, short, direct and naturally adapted to the doubling structure of the Sharkovsky ordering.

A Sharkovsky Theorem for Vertex Maps on Trees

Let T be a tree with n vertices. Let f : T → T be continuous and suppose that the n vertices form a periodic orbit under f . We show: (1) (a) If n is not a divisor of 2k then f has a periodic point with period 2k. (b) If n = 2pq, where q > 1 is odd and p ≥ 0, then f has a periodic point with period 2pr for any r ≥ q. (c) The map f also has periodic orbits of any period m where m can be obtained...

Sharkovsky’s Theorem

We give a proof of Sharkovsky’s theorem that illuminates the doubling structure of the Sharkovsky ordering and that uses the expository device of divide-and-conquer: It breaks into two main parts, each of which again breaks into two parts.

Co-ordering and Type 2 co-ordering

In [arXiv:1006.4939] the enumeration order reducibility is defined on natural numbers. For a c.e. set A, [A] denoted the class of all subsets of natural numbers which are co-order with A. In definition 5 we redefine co-ordering for rational numbers. One of the main questions there, was:"For a specific c.e. set A, consider set of all enumerations of it which is generated by some Turing machine {...