CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 6, Pages 1–12 (January 16, 2002) S 1088-4173(02)00080-2 INDECOMPOSABLE CONTINUA IN EXPONENTIAL DYNAMICS ROBERT L. DEVANEY AND XAVIER JARQUE Abstract. In this paper we prove the existence of uncountably many inde- composable continua in the dynamics of complex exponentials of the form E λ (z)= λe z with λ> 1/e. These continua contain points that share the same itinerary under iteration of E λ . These itineraries are bounded but con- sist of blocks of 0’s whose lengths increase, and hence these continua are never periodic. 1. Introduction Our goal in this paper is to discuss the set of points that share the same itinerary under iteration of complex exponential functions of the form E λ (z )= λe z , where λ> 0. For the moment suppose that λ = 1 so that we consider the usual exponential function E(z )= e z . It is known [14] that the Julia set of E, J (E), is the entire complex plane. Hence E is chaotic everywhere in C. As in [6] we may use symbolic dynamics to describe the fates of orbits of E. We partition the plane into horizontal strips R j of height 2π and centered about the line Im z =2jπ. We may then assign an infinite sequence of integers S(z )= s 0 s 1 s 2 ... to each z via the rule s j = k iff E j (z ) ∈ R k . S(z ) is called the itinerary of z . We make this definition more precise in Section 2 below. A natural question in dynamics is to determine the set of points whose orbits share the same itinerary. A number of results are known in this context for E. For example, if S(z ) is a bounded sequence that consists of at most finitely many zeroes, then the set of points that share this itinerary is a continuous curve homeomorphic to the half line [0, ∞) and extending to ∞ in the right half plane. These curves are called hairs. All orbits on this curve (except possibly that of the endpoint) tend to ∞ in the right half plane [6]. This phenomenon occurs for all exponentials [2]. A contrasting case occurs for the itinerary that is identically zero. In this case the set of points having this itinerary is a pair of invariant indecomposable continua [4], [13]. An indecomposable continuum is a compact, connected subset of the Riemann sphere that cannot be written as a union of two such compact, connected sets. The prototype for such a set is the Knaster continuum [11], [10]. These objects appear often in real dynamical systems [1]. The dynamics on this continuum are quite simple. There is a unique repelling fixed point which is the α-limit set for all orbits in the continuum. All other orbits either tend to ∞ or else accumulate on both Received by the editors August 29, 2001 and, in revised form, November 24, 2001. 2000 Mathematics Subject Classification. Primary 37F10. c 2002 American Mathematical Society 1