Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity

We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard’s celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is sufficient, namely that every point in the bifurcation locus is the landing point of a parameter ray. For exponential maps, the bifurcation locus is not locally connected. We describe a different conjecture (triviality of fibers) which naturally generalizes the role that local connectivity has for quadratic or unicritical polynomials. 1. Bifurcation Loci and Stable Components The family of quadratic polynomials pc : z 7→ z 2 + c, parametrized by c ∈ C, contains, up to conformal conjugacy, exactly those polynomials with only a single, simple, critical value (at c). Hence this family is the simplest parameter space in the dynamical study of polynomials, and has correspondingly received much attention during the last two decades. Similarly, exponential maps Ec : z 7→ e z + c are, up to conformal conjugacy, the only transcendental entire functions with only one singular value (the asymptotic value at c). This simplest transcendental parameter space has likewise been studied intensively since the 1980s. In the following, we will treat these parameter spaces in parallel, unless explicitly stated otherwise; we will write fc for pc or Ec. Following Milnor, we write f ◦n c for the 2000 Mathematics Subject Classification. 37F45 (primary), 30D05, 37F10, 37F20 (secondary).