Path Representation of One-Dimensional Random Walks

Abstract

GL(V) for V a complex vector space [7]. Then there is the fascinating theory of iterations of rational functions. Ours are the quotients of linear polynomials and thus rather simple, but if one considers quotients of polynomials of degree greater than or equal to 2, one begins to venture into the rich area of complex dynamics (see Beardon [1], and also Devaney [2] for general analytic functions). As a single example, one can show that any rational function of degree 5 in numerator and denominator (even with just integer coefficients!) has periodic points (under iteration) of all orders (see Beardon [1, Theorem 6.2.2]). Finally, there are a few open questions suggested by this article, such as: for which algebraic number fields K will all isomorphic finite groups in PGL(2, K) actually be conjugate in PGL(2, K)? This is certainly true for K = C (see Shurman [8], and also Theorem 2.6.1 in the paper by Lyndon and Ullman [5]), and certainly not true for K = Q (as seen in the remark following Lemma 1, above). Most likely, more can be said. One might also ask how many nonconjugate groups (isomorphic to D3, say) are in PGL(2, Q), and if there is a way to describe or index them all.