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For almost every tent map, the turning point is typical. (English) Zbl 0962.37015
Summary: Let \(T_a\) be the tent map with slope \(a\). Let \(c\) be its turning point, and \(\mu_a\) the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function \(g\), it is shown that for almost every \(a\), \(\int g d\mu_a= \lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} g(T_a^i(c))\). As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

MSC:
37E05 Dynamical systems involving maps of the interval
37A05 Dynamical aspects of measure-preserving transformations
28D20 Entropy and other invariants
37A30 Ergodic theorems, spectral theory, Markov operators
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