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We will prove that, for a chaotic mapping f belonging to a suitable class of $C^{1 + \alpha } $ functions, the set $A_\infty ( f )$ has Lebesgue measure zero, with $A_\infty ( f )$ a nonempty set consisting of points whose orbits do not converge to an asymptotically stable periodic orbit of f or to the absorbing boundary. Moreover almost every point is asymptotically periodic with period p, for some positive integer p.Further, we will show that the same conclusions hold for maps with nonpositive Schwarzian derivative under some additional assumptions.

Asymptotically Periodic Behaviour in the Dynamics of Chaotic Mappings