For almost every tent map, the turning point is typical

Volume 155 / 1998

Henk Bruin Fundamenta Mathematicae 155 (1998), 215-235 DOI: 10.4064/fm-155-3-215-235


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, $ʃ g dμ_a = lim_{n → ∞} \frac1n ∑_{i=0}^{n-1} g(T^i_a(c))$. 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.


  • Henk Bruin

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