Circle maps with a flat spot are studied which are differentiable, even on the boundary of the flat spot. Estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set are obtained. Also, a sharp transition is found from degenerate geometry similar to what was found earlier for non-differentiable maps with a flat spot to bounded geometry as in critical maps without a flat spot.

The group $SL(n,{\bf Z})$ acts linearly on $\mathcal{R}^n$, preserving the integer lattice $\mathcal{Z}^{n} \subset \mathcal{R}^{n}$. The induced (left) action on the n-torus $\mathcal{T}^{n} = \mathcal{R}^{n}/\mathcal{Z}^{n}$ will be referred to as the ``standard action''. It has recently been shown that the standard action of $SL(n,\mathcal{Z})$ on $\mathcal{T}^n$, for $n \geq 3$, is both topologically and smoothly rigid. That is, nearby actions in the space of representations of $SL(n,\mathcal{Z})$ into ${\rm Diff}^{+}(\mathcal{T}^{n})$ are smoothly conjugate to the standard action. In fact, this rigidity persists for the standard action of a subgroup of finite index. On the other hand, while the $\mathcal{Z}$ action on $\mathcal{T}^{n}$ defined by a single hyperbolic element of $SL(n,\mathcal{Z})$ is topologically rigid, an infinite dimensional space of smooth conjugacy classes occur in a neighborhood of the linear action. The standard action of $SL(2, \mathcal{Z})$ on $\mathcal{T}^2$ forms an intermediate case, with different rigidity properties from either extreme. One can construct continuous deformations of the standard action to obtain an (arbritrarily near) action to which it is not topologically conjugate. The purpose of the present paper is to show that if a nearby action, or more generally, an action with some mild Anosov properties, is conjugate to the standard action of $SL(2, \mathcal{Z})$ on $\mathcal{T}^2$ by a homeomorphism $h$, then $h$ is smooth. In fact, it will be shown that this rigidity holds for any non-cyclic subgroup of $SL(2, \mathcal{Z})$.

Consider $d$ disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated to the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of $C^1$ conjugation. We solve the inverse problem posed by Dennis Sullivan: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it.

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then we show that the measure $\mu$ has a local product structure and that the currents $\mu^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $\mu$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems.