## 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

### Citations:

Zbl 0664.58015; Zbl 0818.58013
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