| For a given unimodal map $F:I\to I$ on the interval $I$, we consider symmetric unimodal maps (models) so that they are conjugate to $F$. The question concerned here 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 a sufficient "reserve of local evenness" at the turning point (in terms of one-sided higher derivatives at the turning point). Then we consider from different points of view the relationship between dynamical and ergodic properties of unimodal maps and of symmetric Lorenz maps. We present a one-to-one correspondence, which preserves 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 $F$ ; where by Lorenz model of $F$ we mean the discontinuous map obtaining from $F$ by reversing its decreasing branch. |