Chebyshev action on finite fields

Given a polynomial ϕ(x)ϕ(x) and a finite field FqFq one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under ϕϕ. When ϕϕ is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and then use these graphs to determine the decomposition of primes in the Chebyshev radical extensions.