Recurrence Relations

Abstract

In the upcoming chapter we introduce recurrence relations. These are equations that define in recursive fashion, via suitable functions, the terms appearing in a real or complex sequence. The first section deals with some well-known examples that show how these relations may arise in real life, e.g., the Lucas Tower game problem or the death or life Titus Flavius Josephus problem. We then devote a large part of the chapter to discrete dynamical systems, namely recurrences of the form $x_{n+1}=f(x_n)$ where f is a real valued function: in this context the sequence that solves the recurrence, starting from an initial datum, is called the orbit of the initial point. We thoroughly study the case where f is monotonic, and the periodic orbits. The last part of the chapter is devoted to the celebrated Sarkovskii theorem, stating that the existence of a periodic orbit of minimum period 3 implies the existence of a periodic orbit of arbitrary minimum period: we thus give to the reader the taste of a chaotic dynamical system, although that notion is not explicitly developed in this book.