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SARKOVSKII ORDERING

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The Sarkovskii ordering of the positive integers is fascinating, not only because of its strangeness but also because of the elementary devices used in its construction. There are two papers here. One is a consolidation of the partial results and proofs first given in a series of published papers (not all of which were available to me, so some of the constructions may be original). The second, which is not really reliant on the first, deals with a quadratic function where all periodic points can be given explicitly in terms of elementary trigonometric functions.

Both papers were delivered at the South Texas Mathematics Consortium in Edinburg, Texas on February 21, 2009. I an indebted to Ben Brink, my colleague at Wharton County Junior College, for his assistance with the organization and preparation of the manuscripts, and participation in the oral presentations.

Because they are symbolically complex and contain hand-drawn illustrations, we have scanned and thumbnailed the pages. Hopefully, you will be able to print the full resolution images if you wish a hard copy of the paper. It is not copyrighted or otherwise privileged, so enjoy this encounter with one of my favorite gems of mathematics!

FINDING ORDER IN CHAOS: A GEM OF MATHEMATICS

Unification and proof of the Sarkovskii theorems on the existence of periodic points of endomorphisms of the real line (or an interval) ...