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Infinite kneading matrices and weighted zeta functions of interval maps. (English) Zbl 0824.58038

Let \(f : I \to I\) be a piecewise continuous, piecewise monotone transformation of the interval \(I\). In J. Milnor and W. Thurston [Lect. Notes Math. 1342, 465-563 (1988; Zbl 0664.58015)] an unweighted zeta function attached to \(f\) is expressed as a determinant of a finite “kneading matrix”. The author and D. Ruelle [Ergodic Theory Dyn. Syst. 14, No. 4, 621-632 (1994; Zbl 0818.58013)]extended this result to the weighted case under the assumption that the weight function \(g : I \to \mathbb{C}\) has to be piecewise constant.
In the present paper this theory is extended to the case of a more general weight which only has to be constant on “homtervals”. These are intervals that are left stable by every subdivision of \(I\) given by higher preimages under \(f\). It may happen that there are no homtervals at all. It turns out that in this extended setting there are no finite kneading matrices giving the weighted zeta function as a determinant but an infinite sequence of kneading matrices whose determinants converge to the zeta function.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37E99 Low-dimensional dynamical systems
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