Ergodic Theory and Dynamical Systems



Decay of geometry in the cubic family


GRZEGORZ SWIATEK  1 a1 and EDSON VARGAS  2 a2
a1 Penn State University, 209 Mc Allister, University Park, PA 16802, USA
a2 IME-USP, Cx. P.: 66281 CEP: 05315-970, São Paulo, Brasil

Abstract

We study the box geometry of bi-modal maps. As has been expected, unlike in the unimodal case, there may be non-renormalizable mappings which do not show the decay of geometry. We provide an example. On the other hand, we come up with a natural pattern in which the critical orbits are closely intertwined, but the decay of geometry persists. All this is done in the context of a generally defined box inducing construction for bi-modal maps.

(Received March 30 1996)
(Revised June 27 1997)



Footnotes

1 Partially supported by NSF Grant #DMS-9704368, Sloan Foundation and FAPESP-Brasil, Grant #94/5959-1.

2 Partially supported by CNPq-Brasil, Grant \#300557/89-2(RN) and #453649/95-5.