Abstract:

Among transcendental functions, the complex exponential family $ez+c$ is one of the most widely studied examples, because there is only one singular value at c and because in some respects it can be seen as combinatorial limit of unicritical polynomials of degree d. We will explore some features of a specific class of parameter in this family, those for whom the singular value $c$ is preperiodic, and describe the combinatorial rigidity conjecture, concluding with a sketch of the proof of rigidity (triviality of fibers) for this 'simple' class of parameters.