Smooth Symmetric and Lorenz Models for unimodal Maps

For a given unimodal map F:I→I on the interval I, we consider symmetric unimodal maps (models) so that they are conjugate to F. The question motivated by [Gambaudo & Tresser, 1992] is the following: whether it is possible for symmetric model to preserve smoothness of the initial map F? We construct a symmetric model which is proved to be as smooth as F provided F has a nonflat turning point with sufficient "reserve of local evenness" at the turning point (in terms of one-sided higher derivatives at the turning point, see Definition 2.4 and Theorem 2.7). We also consider from different points of view the relationship between dynamical and ergodic properties of unimodal maps and of symmetric Lorenz maps. In particular, we present a one-to-one correspondence preserving the measure theoretic entropy, between the set of invariant measures of a symmetric unimodal map F and the set of symmetric invariant measures of the Lorenz model of F (Theorem 3.5), where by Lorenz model of F we mean the discontinuous map obtained from F by reversing its decreasing branch. Finally we extend for nonsymmetric unimodal maps, the result of Gambaudo and Tresser [1992] on Ck structural instability of the maps whose rotation interval has irrational end point (answering a question from [Gambaudo & Tresser, 1992]).