Abstract

I will give a brief summary of the existence of indecomposable continua (that is, a compact connected space that cannot be expressed as the union of two compact connected spaces) in the framework of complex exponentials. I will show (among other results) the existence of uncountably many indecomposable continua in the dynamics of complex exponentials of the form $E$_λ(z) = λ e^z with $\lambda \; >\; 1/e$. These continua contain points that share the same itinerary under iteration of $E$_λ. These itineraries are bounded but consist of blocks of $0$´s whose lengths increase, and hence these continua are never periodic. I will also deal with the Misiurewicz case where it seems that more complicated scenario can be observe.