Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

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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: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (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.

\S 1. IntroductionRecently codimension 3 embeddings have been discussed in many papers. It is known that codimension 3 topological embeddings are approximated by PL embeddings and close (topological or PL) embeddings are isotopic (e.g. [2]).In this paper we shall compare homotopy groups of s.s. complexes of topological embeddings with those of PL embeddings, and show that the local contractibility of spaces of topological embedding implies similar properties of PL embeddings with codimension \geqq 3 .Main results are followings.. THEOREM A. Suppose that M^{m} , Q^{q} are m-dim. and q-dim. PL manifolds, and \mathcal{E}^{TOP}(M, Q) (resp. \mathcal{E}^{PL}(M, Q) ) is a Kan complex of locally flat topological (resp. PL) embeddings. Then 1) If q-m\leqq 2 q\geqq 5 , \mathcal{E}^{PL}(M, Q) is homotopy equivalent to \mathcal{E}^{TOP}(M, Q) provided H^{i}(M, Z_{2})=0 for i\leqq 3 .2) If q-m\geqq 3 or q\leqq 3 , \mathcal{E}^{PL}(M, Q) is homotopy equivlent to \mathcal{E}^{T0P}(M , Q)THEOREM B. When q-m\geqq 3, \mathcal{E}^{PL}(M, Q) is locally n-connected for any n\geqq 0 . This gives a partial answer to the question of Edward ([2]). The case when n\leqq q-m-3 is obtained by Lusk ([8]).By rB^{k} and rS^{k-1} we denote the k-ball with radius r\{(X_{1}, \cdots, X_{k})\in R^{k}||X_{i}|\leqq r\} and the (k-1)-sphere\partial(rB^{k}) , respectively. R^{k} is identified with the subspace R^{k}\cross 0\subset R^{k+1} . ASSUMPTIONS. In this paper we assume the followings: A map denoted by f:M\cross\Delta^{k}arrow Q\cross\Delta^{k} is level preserving i.e. pr_{2}\circ f=pr_{2} , f_{t} is given by f(x, t)=(f_{t}(x), t) , and f(M\cross\Delta^{k})\subset intQ\cross\Delta^{k} . It is valid to generalize the result in this paper for proper embeddings. Embeddings and immersions are to be locally flat.For manifolds M^{m} and Q^{q} , we assume m