Invariant measures exist under a summability condition for unimodal maps

SummaryFor unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.

[1]  M. Lyubich,et al.  Attractors of maps of the interval , 1989 .

[2]  Stewart D. Johnson.  Singular measures without restrictive intervals , 1987 .

[3]  S. Strien.  On the bifurcations creating horseshoes , 1981 .

[4]  Alexander Blokh,et al.  Measurable dynamics of $S$-unimodal maps of the interval , 1991 .

[5]  L. Young,et al.  Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps , 1992, Ergodic Theory and Dynamical Systems.

[6]  S. Strien.  Hyperbolicity and invariant measures for generalC2 interval maps satisfying the Misiurewicz condition , 1990 .

[7]  M. Lyubich,et al.  Attractors of transformations of an interval , 1987 .

[8]  Gerhard Keller,et al.  Exponents, attractors and Hopf decompositions for interval maps , 1990, Ergodic Theory and Dynamical Systems.

[9]  Symmetric S-unimodal mappings and positive Liapunov exponents , 1985, Ergodic Theory and Dynamical Systems.

[10]  M. Jakobson.  Absolutely continuous invariant measures for one-parameter families of one-dimensional maps , 1981 .

[11]  Pierre Collet,et al.  Positive Liapunov exponents and absolute continuity for maps of the interval , 1983, Ergodic Theory and Dynamical Systems.

[12]  S. Strien,et al.  Absolutely continuous invariant measures forC2 unimodal maps satisfying the Collet-Eckmann conditions , 1988 .

[13]  A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity , 1988, Ergodic Theory and Dynamical Systems.

[14]  J. Yorke,et al.  On the existence of invariant measures for piecewise monotonic transformations , 1973 .

[15]  S. Strien,et al.  A structure theorem in one dimensional dynamics , 1989 .

[16]  John Guckenheimer,et al.  Limit sets ofS-unimodal maps with zero entropy , 1987 .

[17]  Michał Misiurewicz,et al.  Absolutely continuous measures for certain maps of an interval , 1981 .

[18]  M. Martens,et al.  Julia-Fatou-Sullivan theory for real one-dimensional dynamics , 1992 .