** Authors:**
JUDY A. KENNEDY, JAMES A. YORKE

** Abstract: **
We first give a precise definition of the terms
``topological horseshoe'' and ``generalized quadrilateral'' and then
examine the behavior of a homeomorphism F on a locally compact,
separable, locally connected metric space X (X is usually a manifold
in applications) such that F restricted to some generalized
quadrilateral Q in X is a topological horseshoe map. For a set
Q \subset X we define and describe (1) the permanent set Z of Q to be
\{x\in X:F^n(x)\in Q for all integers n\}, and (2) the entrainment set
of Q to be E(Q)=\{x\in X: F^{-n} (x)\in Q for all sufficiently large
n\}. We give conditions under which various closed sets of
\overline{E(Q)} are associated, in a strong way, with indecomposable,
closed, connected spaces invariant under F. (A connected set A is
indecomposable if it is not the union of two proper connected sets,
each of which is closed relative to A.) Next we show that even when
small amounts of noise are added to the dynamical system, there are
associated indecomposable sets. These sets are not, in general,
invariant sets for our process with noise, but they are the physically
observable sets, while invariant Cantor sets are not, and they are the
sets that can be measured.

** Keywords: **
topological horseshoes, generalized
quadrilateral, indecomposable continua, connected indecomposable
sets, shift dynamics, noisy dynamical systems, permanent set,
entrainment set, destination set.

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