کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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6424922 | 1633784 | 2017 | 16 صفحه PDF | دانلود رایگان |
We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if X is an arc-like continuum which admits a homeomorphism f with positive topological entropy, then X contains an indecomposable subcontinuum. Barge and Diamond proved that if G is a finite graph and f:GâG is any map with positive topological entropy, then the inverse limit space limâ(G,f) contains an indecomposable continuum. In this paper we show that if X is a G-like continuum for some finite graph G and f:XâX is any map with positive topological entropy, then limâ(X,f) contains an indecomposable continuum. As a corollary, we obtain that in the case that f is a homeomorphism, X contains an indecomposable continuum. Moreover, if f has uniformly positive upper entropy, then X is an indecomposable continuum. Our results answer some questions raised by Mouron and generalize the above mentioned work of Mouron and also that of Barge and Diamond. We also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result.
Journal: Advances in Mathematics - Volume 304, 2 January 2017, Pages 793-808