The dynamics of continuous maps of finite graphs through inverse limits

Barge, Marcy; Diamond, Beverly

doi:10.1090/S0002-9947-1994-1236222-1

The dynamics of continuous maps of finite graphs through inverse limits
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by Marcy Barge and Beverly Diamond

Trans. Amer. Math. Soc. 344 (1994), 773-790

DOI: https://doi.org/10.1090/S0002-9947-1994-1236222-1

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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 \[ {I_1} \cup {I_2} \subseteq {f^n}({I_1}) \cap {f^n}({I_2});\] (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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