Infinite-modal maps with global chaotic behavior. (English) Zbl 0916.58029

Pacifico, M. J.; Rovella, A.; Viana, M.

Ann. Math. (2) 148, No. 2, 441-484 (1998); Corrigendum 149, No. 2, 705 (1999).

The main goal is to study the dynamics of certain parametrized families \((f_\mu)_\mu\) of one-dimensional maps with infinite critical set. The authors show that they exhibit global chaotic behaviour in a persistent way: For a positive Lebesgue measure set of parameter values the map is transitive and almost every orbit has positive Lyapunov exponent. This class of systems is motivated by a problem in the dynamics of flows in three dimensions: the unfolding of saddle-focus homoclinic connections.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37D45

Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

critical set; chaotic behaviour; homoclinic connections; saddle-focus homoclinic connections