Chris Bernhardt

Simple permutations with order a power of two

Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This paper studies the structure of simple permutations which have order a power of two, where simple permutations are permutations corresponding to the simple orbits of Block.

The ordering on permutations induced by continuous maps of the real line

Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied. Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.

A Proof of Sharkovsky's Theorem

This paper presents a variation on the standard directed graph proof. This variation makes use of the natural representation of the symmetric group given by one-dimensional maps.

Periodic orbits of continuous mappings of the circle without fixed points

Let f be a continuous map of the circle to itself. Let P(f) denote the set of periods of the periodic points. In this paper the set P(f) is studied for functions without fixed points, so 1∉ P(f) . In particular, it is shown that if s, t are the two smallest integers in P(f) and s and t are relatively prime then α s +β t ∈ P(f) for any positive integers α and β.

A Sharkovsky theorem for vertex maps on trees

Let T be a tree with n vertices. Let 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 2 k then f has a periodic point with period 2 k . b. If , where is odd and , then f has a periodic point with period 2 p r for any . c. The map f also has periodic orbits of any period m where m can be obtained from n by removing ones from the right of the binary expansion of n and changing them to zeroes. 2. Conversely, given any n, there is a tree with n vertices and a map f such that the vertices form a periodic orbit and f has no other periods apart from the ones given above.

Vertex maps on graphs-trace theorems

AbstractThe paper proves two theorems concerning the traces of Oriented Markov Matrices of vertex maps on graphs. These are then used to give a Sharkoksky-type result for maps that are homotopic to the identity and that flip at least one edge. 2000 Mathematics Subject Classification 37E15, 37E25

Rotation matrices for vertex maps on graphs

Let G be a finite connected graph. Suppose is a map homotopic to the identity that permutes the vertices. For such a map, a rotation matrix is defined and the basic properties of this matrix are given. It is shown that this matrix generalizes some of the information given by the rotation interval, which is defined when the graph is a circle, to more general graphs.

Periods of orbits for maps on graphs homotopic to a constant map

This paper proves two theorems concerning the set of periods of periodic orbits for maps of graphs that are homotopic to the constant map and such that the vertices form a periodic orbit. The first result is that if the number of vertices is not a divisor of 2 k then there must be a periodic point with period 2 k . The second is that if the number of vertices is for odd s>1, then for all r>s there exists a periodic point of minimum period . These results are then compared to the Sharkovsky ordering of the positive integers.

A Sharkovsky Theorem for a Discrete System

A class of discrete systems is described. It is shown that if one of these systems has a boundary consisting of periodic point of period n, then the system has a periodic point of period m for all where < s is the ordering of the positive integers given by Sharkovskys Theorem.

A Proof of the Cayley-Hamilton Theorem

Let M (n, n) be the set of all n × n matrices over a commutative ring with identity. Then the Cayley Hamilton Theorem states: Theorem. Let A ∈ M (n, n) with characteristic polynomial det(tI − A) = c 0 t n + c 1 t n−1 + c 2 t n−2 + · · · + c n. Then c 0 A n + c 1 A n−1 + c 2 A n−2 + · · · + c n I = 0. In this note we give a variation on a standard proof (see [1], for example). The idea is to use formal power series to slightly simplify the argument. Proof. First, observe that det (I − tA) = t n det 1 t I − A = c 0 + c 1 t + c 2 t 2 + · · · + c n t n .