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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
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In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.
We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on $T^2$ which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
We discuss some properties of a class of cellular automata sometimes called a "generalized ant". This system is perhaps most easily understood by thinking of an ant which moves about a lattice in the plane. At each vertex (or "cell"), the ant turns right or left, depending on the the state of the cell, and then changes the state of the cell according to certain prescribed rule strings. (This system has been the subject of several Mathematical Entertainments columns in the Mathematical Intelligencer; this article will be a future such column). At various times, the distributions of the states of the cells for certain ants is bilaterally symmetric; we categorize a class of ants for which this is the case and give a proof using Truchet tiles.
It is shown that a polynomial with a Cremer periodic point has a non-accessible critical point in its Julia set provided that the Cremer periodic point is approximated by small cycles.
Consider the group $Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\omega)$ with the Hofer $L^{\infty}$-norm. A path in $Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\mathcal{L}$. In this paper, we give a necessary condition for a path $\gamma$ to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for $\mathcal{L}$. This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky's work on the second variation formula. Using it, we construct a symplectomorphism of $S^2$ which cannot be reached from the identity by a shortest path. In later papers in this series, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We will also investigate conditions under which geodesics are absolutely length-minimizing.
In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on $M$. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic to $M \times D^2$ which are symplectically ruled over $D^2$. When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided that $M$ is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory of $J$-holomorphic curves in arbitrary $M$.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongst {\it all} paths, not only the homotopic ones) under even more restrictive conditions on $M$, for example when $M$ is exact and convex or of dimension $2$. The new difficulty is caused by the possibility that there are non-trivial and very short loops in $Ham^c(M)$. When such length-minimizing paths do exist, we can extend the Bialy--Polterovich calculation of the Hofer norm on a neighbourhood of the identity ($C^1$-flatness).
Although it applies to a more restricted class of manifolds, the Hofer-Zehnder capacity seems to be better adapted to the problem at hand, giving sharper estimates in many situations. Also the capacity-area inequality for split cylinders extends more easily to quasi-cylinders in this case. As applications, we generalise Hofer's estimate of the time for which an autonomous flow is length-minimizing to some manifolds other than $\textbf{R}^{2n}$, and derive new results such as the unboundedness of Hofer's metric on some closed manifolds, and a linear rigidity result.
Irreducible 3-manifolds are divided into Haken manifolds and non-Haken manifolds. Much is known about the Haken manifolds and this knowledge has been obtained by using the fact that they contain incompressible surfaces. On the other hand, little is known about non-Haken manifolds. As we cannot make use of incompressible surfaces we are forced to consider other methods for studying these manifolds. For example, exploiting the structure of their Heegaard splittings. This approach is enhanced by the result of Casson and Gordon [CG1] that irreducible Heegaard splittings are either strongly irreducible (see Definition 1.2) or the manifold is Haken. Hence, the study of Heegaard splittings as a mean of understanding 3-manifolds, whether they are Haken or not, takes on a new significance.
One of the main questions in the field of complex dynamics is the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected. In this paper we shall prove the following
Main Theorem: Let $f$ be a polynomial of the form $f(z)=z^d +c$ with $d$ an even integer and $c$ real. Then the Julia set of $f$ is either totally disconnected or locally connected. In particular, the Julia set of $z^2+c$ is locally connected if $c \in [-2,1/4]$ and totally disconnected otherwise.
We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, with a left invariant bracket-generating distribution, for which some Carnot geodesics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explicitly in terms of the structure constants.
The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one of us (Sullivan) gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than Sullivan's Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria.