## Institute for Mathematical Sciences

## Preprint ims90-9

** L. Chen **
* Shadowing Property for Nondegenerate Zero Entropy Piecewise Monotone Maps*

Abstract: Let *f* be a continuous piecewise monotone map of the interval. If any two periodic orbits of *f* have different itineraries with respect to the partition of the turning points of *f*, then *f* is referred to as "nondegenerate". In this paper we prove that a nondegenerate zero entropy continuous piecewise monotone map *f* has the Shadowing Property if and only if 1) *f*dows not have neutral periodic points; 2) for each turning point *c* of *f*, either the ω-limit set ω(*c*,*f*) of *c* contains no periodic repellors or every periodic repellor in ω(*c*,*f*) is a turning point of *f* in the orbit of *c*. As an application of this result, the Shadowing Property for the Feigenbaum map is proven.

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