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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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