Implicit formalism for affine-like maps and parabolic composition (Q2770228)

Affine-like maps and parabolic maps, which play a fundamental role in the analysis of nonuniformly hyperbolic dynamics on surfaces, are presented. Affine-like maps occur in a natural way when looking for basic sets through Markov partitions. To deal with these maps, a time-symmetric implicit formalism is used. The main technical tool is distortion, that measures the deviation from affine maps and must be scaled in an appropriate way. Parabolic maps are related to the quadratic tangency which occurs at the homoclinic bifurcation. To study the full dynamics, one needs to compose parabolic maps with affine-like maps and to know under which circumstances one again obtains an affine-like map. This is only possible if a transversability condition is satisfied. The main result of the paper is an estimate on the distortion of the resulting maps under the transversality hypothesis.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00062].