Dynamical properties of maps derived from maps with strong negative Schwarzian derivative

Statistical Properties of Unimodal Maps: Smooth Families with Negative Schwarzian Derivative

We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves the Palis conjecture in this ...

On Iterated Positive Schwarzian derivative Maps

The families of one-dimensional maps have been the object of many studies since [Sharkovsky, 1964], for instance in [Sharkovsky et al., 1997] or in [de Melo & van Strien, 1993], whose authors also studied this subject, we have two large surveys of the enormous effort made in the last decades in this field of work. We adopt the usual definitions as for instance symbolic dynamics or topological e...

Real Bounds, Ergodicity and Negative Schwarzian for Multimodal Maps

Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained s...

Schwarzian Derivatives and Cylinder Maps

We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval I affects the dynamics of an associated many-to-one skew product map of the cylinder (R/Z) × I.

Statistical properties of intermittent maps with unbounded derivative