| Consider the family of complex exponentials Elz=l ez . In [Dev93], Devaney showed that for any real parameter l > 1/e, the Julia set of El contains an invariant set that can be countably compactified into an indecomposable continuum. Given any two parameters l , mu > 1/e, l ≠ mu, the maps El and Emu are not topologically conjugate [DG88]. It has been conjectured by Devaney that the indecomposable continua associated to each map are not homeomorphic.;In the present work, two main results will be introduced. First, I will prove the existence of indecomposable continua for certain complex parameters. Then, I will present significant results towards the solution of Devaney's conjecture for the real parameter case.;The generalization will be provided for complex parameters for which the orbit of zero lies on n dynamical curves and tends to infinity. Hence, those values of l lie in the unstable region of the parameter plane and the Julia set is the whole complex plane (after [Sul85] and [GK86]). I will show the existence of fundamental domains and the existence of corresponding invariant sets under Enl inside each domain. Their compactification will result in continua with the same properties as the one described in [Dev93].;To address the conjecture, I will introduce a model in the plane that has dynamics similar to that of the complex exponential family. This model is based on a one parameter family of continuous, piecewise semilinear maps, hl , with l > 1. Under certain assumptions on the parameter value, I will show that the continuum obtained from the semilinear model has the same topological characteristics as the one obtained for El . Because of the semilinearity, the model is significantly easier to work with in many respects. As an example, each map hl has an associated kneading sequence which is a topological invariant for the map. The computation of the kneading invariant will show that, for different parameters l and mu, the respective maps are not topologically conjugate, despite the fact that their gross dynamical properties are the same. This will be an analogus result to the one given in [DG88] for the exponential family. |
