AI Summary & Key Takeaways
AI InsightThese 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.
📌 3 Key Takeaways
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:Fn(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.
Özet
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:Fn(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.