Hyperbolicity and Invariant Measures for General (C sup 2) Interval Maps Satisfying the Misiurewicz Condition

摘要

The paper shows that C sup 2 mappings on (0,1) or S sup 1 satisfying the so-called Misiurewicz conditions are globally expanding (in the sense defined below) and have absolute continuous invariant measures of positive entropy. Assumptions are not needed on the Schwarzian derivative of these maps. Instead it is necessary that all critical points of f are non-flat, and that f has no periodic attractors. The proof gives an algorithm to verify the last condition. The result implies the result of Misiurewicz (where only maps with negative Schwarzian derivatives are considered). Moreover, as a byproduct, the present paper implies (and simplifies the proof of) the results of Mane, who considers general C sup 2 maps (without conditions on the Schwarzian derivative), and restricts attention to points whose forward orbits stay away from the critical points. One of the main complications will be that in this case it is necessary to consider points whose iterations come arbitrarily close to critical points. In order to deal with this non-linearity the tools of PB87-225984 are used. It will turn out that the estimates obtained are so precise that invariant measures can be proved in a very simple way. (Copyright (c) 1987 by Faculty of Mathematics and Informatics, Delft, The Netherlands.)