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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The dynamics of continuous maps of finite graphs through inverse limits


Authors: Marcy Barge and Beverly Diamond
Journal: Trans. Amer. Math. Soc. 344 (1994), 773-790
MSC: Primary 58F03; Secondary 54H20, 58F13
MathSciNet review: 1236222
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Abstract: Suppose that $ f:G \to G$ is a continuous piecewise monotone function on a finite graph G. Then the following are equivalent:

(i) f has positive topological entropy;

(ii) there are disjoint intervals $ {I_1}$, and $ {I_2}$ and a positive integer n with

$\displaystyle {I_1} \cup {I_2} \subseteq {f^n}({I_1}) \cap {f^n}({I_2});$

(iii) the inverse limit space constructed by using f on G as a single bonding map contains an indecomposable subcontinuum.

This result generalizes known results for the interval and circle.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1236222-1
PII: S 0002-9947(1994)1236222-1
Keywords: Inverse limit, indecomposable continuum, finite graph, topological entropy, horseshoe
Article copyright: © Copyright 1994 American Mathematical Society