Indecomposable Continua and Misiurewicz Points in Exponential Dynamics Robert L. Devaney * Mathematics Department Boston University Boston, MA 02215, USA Xavier Jarque † Dep. Mat. Aplicada i An`alisi Universitat de Barcelona Gran Via 585 08007 Barcelona, Spain M´onica Moreno Rocha ‡ Mathematics Department Tufts University Medford, MA 02155, USA January 19, 2005 1 Introduction In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions E λ (z )= λe z where λ ∈ C. These invariant sets consist of points that share the same itinerary under iteration of E λ . Since these exponential functions are 2πi periodic, there are several “natural” ways (described below) to decompose the plane into countably many strips of vertical height 2π which are then indexed in the natural way by the integers according to the increasing imaginary parts of the strips. The itinerary of a point is then the sequence of integers that describes how the orbit of that point passes through these various strips. Thus we investigate the sets of points whose orbits make the transit through these strips in the same fashion. For complex analytic maps, the Julia set consists of all points at which the family of iterates of the map fails to form a normal family in the sense of Montel. Equivalently, the Julia set may be described as either the closure of the set of repelling periodic points or else as the set of points on which the map behaves chaotically. For E λ , the Julia set is also the closure of the set of points whose orbits tend to ∞ [17]. We denote the Julia set of the exponential map by J (E λ ). It is well known that, if J (E λ ) contains an open set, then in fact the Julia set must be the entire plane. Otherwise, J (E λ ) is a nowhere dense subset of the plane. * Partially supported by NSF Grant 02-05779. † Partial support from the Spanish Ministry of education and Science and FEDER through the grants BFM2002– 01344 and SEC2003-00306 and from the Catalan Government through the grant SGR99-00349 is gratefully acknowl- edge. ‡ Supported by FRAC Grant, 2003, Tufts University. 1