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**PREPRINTS IN THIS SERIES, IN PDF FORMAT.**

* Starred papers have appeared in the journal cited.

*On Aubry Mather Sets*

Let f be a two dimensional area preserving twist map. For each irrational rotation number in a certain (non trivial) interval, there is an f-invariant minimal set which preserves order with respect to that rotation number. For large nonlinearity these sets are, typically, Cantor sets and they are referred to as Aubry Mather sets. We prove that under csome assumptions these sets are ordered vertically according to ascending rotation number ("Monotonicity"). Furthermore, if f statisfies certain conditions, the right hand points of the gaps in an irrational Cantor set lie on a single orbit ("Single Gap") and diffusion through these Aubry Mather sets can be understood as a limit of resonance overlaps (Convergence of Turnstiles). These conditions essentially establish the existence of a hyperbolic structure and limit the number of homoclinic minimizing orbits. Some other results along similar lines are given, such as the continuity at irrational rotation numbers of the Lyapunov exponent on Aubry Mather sets.

*Dynamical Properties of Some Classes of Entire Functions*

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is proven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.

*Dynamics in One Complex Variable: Introductory Lectures*

These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 1989-90. These lectures are intended to introduce the reader to some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry.

*Remarks on Iterated Cubic Maps*

This note will discuss the dynamics of iterated cubic maps from the real or complex line to itself, and will describe the geography of the parameter space for such maps. It is a rough survey with few precise statements or proofs, and depends strongly on work by Douady, Hubbard, Branner and Rees.

*Intersection Properties of Invariant Manifolds in Certain Twist Maps*

We consider the space N of $C^2$ twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constant k (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps in N with nonlinearity k large enough.

The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).

In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem).

Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.

*Scalings in Circle Maps (I)*

Let f be a flat spot circle map with irrational rotation number. Located at the edges of the flat spot are non-flat critical points $(S: x \rightarrow Ax^v ,v \geq 1)$. First, we define scalings associated with the closest returns of the orbit of the critical point. Under the assumption that these scalings go to zero, we prove that the derivative of long iterates of the critical value can be expressed in the scalings. The asymptotic behavior of the derivatives and the scalings can then be calculated. We concentrate on the cases for which one can prove the above assumption. In particular, let one of the singularities be linear. These maps arise for example as the lower bound of the non-decreasing truncations of non-invertible bimodal circle maps. It follows that the derivatives grow at a sub-exponential rate.

*Shadowing Property for Nondegenerate Zero Entropy Piecewise Monotone Maps*

Let *f* be a continuous piecewise monotone map of the interval. If any two periodic orbits of *f* have different itineraries with respect to the partition of the turning points of *f*, then *f* is referred to as "nondegenerate". In this paper we prove that a nondegenerate zero entropy continuous piecewise monotone map *f* has the Shadowing Property if and only if 1) *f*dows not have neutral periodic points; 2) for each turning point *c* of *f*, either the ω-limit set ω(*c*,*f*) of *c* contains no periodic repellors or every periodic repellor in ω(*c*,*f*) is a turning point of *f* in the orbit of *c*. As an application of this result, the Shadowing Property for the Feigenbaum map is proven.

*One-Dimensional Maps and Poincare Metric*

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps. (AMS subject code 26A18)

*Scalings in Circle Maps II*

In this paper we consider one parameter families of circle maps with nonlinear flat spot singularities. Such circle maps were studied in [Circles I] where in particular we studied the geometry of closest returns to the critical interval for irrational rotation numbers of constant type. In this paper we apply those results to obtain exact relations between scalings in the parameter space to dynamical scalings near parameter values where the rotation number is the golden mean. Then results on [Circles I] can be used to compute the scalings in the parameter space. As far as we are aware, this constitutes the first case in which parameter scalings can be rigorously computed in the presence of highly nonlinear (and nonhyperbolic) dynamics.

*The Julia Sets and Complex Singularities in Hierarchical Ising Models*

We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known that the singularities of free energy of this model lie on the Julia set of some rational endomorphism $f$ related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin of $f$. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it.