Digest of Master's thesis

TITLE: The continuity of the topological entropy of one dimensional maps with degenerate critical points.

One dimensional maps with degenerate critical points are studied and we give a larger class of C^r functions whose topological entropy is continuous.

There are many studies of topological entropy of one dimensional maps. M. Misiurewicz and W. Szlenk showed that topological entropy is lower semi-continuous for piecewise monotone C⁰ maps with C⁰ topology. Moreover, if none of the critical points of a C¹ map are degenerate, its entropy is continuous with C¹ topology. But when critical points are degenerate, there is a counter example of C^r (r >= 1) functions whose topological entropy is not continuous.

Our aim in this paper is to show the continuity of topological entropy for maps with finitely degenerate critical points. In this paper I means an interval [0,1].

Theorem

Let r >=2 and F^r be a set of C^r functions defined as follows:

F^r= {f of C^r(I,I); for all x in I, there is k, 1 < k <= r such that f^(k)(x) <> 0 }.

Then the topological entropy is continuous in F^r.

To study topological entropy of one dimensional maps, the kneading theory due to J.Milnor and W.Thurston is useful. But when degeneration occurs, we can not apply this theory directly because the lap number changes.

There are essentially two types of degeneration and every degeneration is written by a combination of these two types. One is a degeneration of critical points into one critical point, and the other is that into a saddle.

We improve the kneading theory to be applied to the functions with degenerate critical points and show that the topological entropy is continuous in F^r.