A generalization of a theorem of marotto

Abstract

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f : R n → R n be a differentiate map of the n -dimensional Euclidean space R n (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior. In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.