IJPAM: Volume 1, No. 4 (2002)

A SIMPLE PROOF THAT HOMOCLINICITY IMPLIES
HORSESHOE FOR CONTINUOUS INTERVAL MAPS

Ming-Chia Li
Dept. of Mathematics
National Changhua University of Education
Changhua, Taiwan 500, REPUBLIC OF CHINA
e-mail: mcli@math.ncue.edu.tw


Abstract.In [#!Smale1967!#] Smale, showed that for diffeomorphisms on high dimensions the existence of a transverse homoclinic point implies the existence of a horseshoe and so guarantees chaotic dynamics. In Theorem 16.5 of [#!Devaney1989!#], Devaney showed the same result holds for $C^1$ maps on the real line with a non-degenerate homoclinic point. In Proposition III.16 of [#!BlockCoppel1992!#], Block and Coppel showed that for a continuous map $f$ on the real line a (possibly degenerate) homoclinic point leads a horseshoe for $f^2$. In this paper, based on an elementary proof, we extend the result of Block and Coppel.

Received: February 26, 2002

AMS Subject Classification: 37E05, 37C25

Key Words and Phrases: homoclinicity, $2$-horseshoe, repelling fixed points

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 1
Issue: 4