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Discrete potential theory for iterated maps of the interval. (English) Zbl 1185.37098

Elaydi, Saber (ed.) et al., Advances in discrete dynamical systems. Proceedings of the 11th international conference on difference equations and applications (ICDEA 06), Kyoto, Japan, July 24–28, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-49-5/hbk). Advanced Studies in Pure Mathematics 53, 121-128 (2009).
Let \(f:I \rightarrow I\) be a piecewise monotone interval map. The singular points of \(f\) are the points \(x \in I\) such that \(f'(x)=0\), \(f'(x)\) does not exist, or \(f'\) is not continuous at \(x\). In this paper, the authors assume that \(f\) has a finite number of singular points and that the kneading sequence of each singular point is eventually periodic (in some sense the orbit of a singular point is finite). First a Markov digraph is derived from a transition matrix determined by \(f\) and the finite orbits of the singular points of \(f\). Then through the Parry measure, the authors generate a weighted digraph and define a current and a potential which turn out to be dynamical invariants. They prove that the current and potential satisfy the general Kirchoff laws. An example is provided using a quadratic map.
For the entire collection see [Zbl 1170.39300].

MSC:

37E05 Dynamical systems involving maps of the interval
31C20 Discrete potential theory
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