Chaotically driven sigmoidal maps

We consider skew product dynamical systems [Formula: see text] with a (generalized) baker transformation [Formula: see text] at the base and uniformly bounded increasing [Formula: see text] fibre maps [Formula: see text] with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for [Formula: see text], we prove that the presence of these fibres restricts considerably the possible structures of invariant measures — both topologically and measure theoretically, and that this finally allows to provide a “thermodynamic formula” for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.

A link between topological entropy and Lyapunov exponents

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

Partially hyperbolic dynamics in dimension three

Partial hyperbolicity appeared in the 1960s as a natural generalization of hyperbolicity. In the last 20 years, there has been great activity in this area. Here we survey the state of the art in some related topics, focusing especially on partial hyperbolicity in dimension three. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of$3$-manifolds influences the dynamics in revealing ways.

Partial hyperbolicity and classification: a survey

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.