Structural Instability of exp(z)

Abstract

The entire function exp(z) has a Julia set eqval to the whole plane. We show that there are complex X's near 1 such that \e: has an attracting periodic orbit. Hence e~ is not structurally stable. The study of the dynamics of complex analytic maps goes back to Fatou and Julia in the 1920s. Recently, there has been a rebirth of interest in this subject, due mainly to the interesting new results of Mandelbrot (Ma), Douady-Hubbard (DH), and Sullivan (S). Most of this recent work has dealt with polynomials or rational maps, rather than with entire maps. The reason for this is that rational maps may be regarded as smooth maps of the Riemann sphere, whereas, because of the essential singularity at infinity, entire maps cannot. Thus, compactness is lost and, with it, many of the global theorems in the subject. Nevertheless, entire maps like ez share many of the dynamical features of rational maps. However, as was shown in (DK), the exponential map has several additional features that are not found in the polynomial case. For example, Misiurewicz (M) has shown that the Julia set of ez is the entire plane, something that cannot occur for polynomials. Recall that the Julia set of a map is the set of points at which the family of iterates of the map fails to be a normal family of functions, and that periodic points are dense in the Julia set (F). In particular, periodic points for ez are dense in the plane. This raises the question of the structural stability of ez. One may check easily that if an entire function is topologically conjugate to ez, then it is affinely equivalent to Xez for some X g C. Hence, structural stability here means within the class of maps Xez. In this limited class of maps, it would seem that there is a good chance for stability. Nevertheless, our main result is that this is not so. Theorem. ez is not structurally stable. We remark that this result is a crucial step in the recent proof of Ghys, Goldberg, and Sullivan (GGS) that ez is recurrent (in the sense that there are no positive measure cross sections to the grand orbits of ez). The natural question of the ergodicity of ez remains open.