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For almost every tent map, the turning point is typical

Henk Bruin (1998)

Fundamenta Mathematicae

Let T a be the tent map with slope a. Let c be its turning point, and μ a the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, ʃ g d μ a = l i m n 1 n 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.

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