We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

Recently, Goldman [2] proved that the mapping class group of a compact surface $S$, ${\it MCG}(S)$, acts ergodically on each symplectic stratum of the Poisson moduli space of flat $ S(2)$-bundles over $S$, $X(S, S(2))$. We show that this property does not extend to that of cyclic subgroups of ${\it MCG}(S)$, for $S$ a punctured torus. The symplectic leaves of $X(T^2-pt., SU(2))$ are topologically copies of the 2-sphere $S^2$, and we view mapping class actions as a continuous family of discrete Hamiltonian dynamical systems on $S^2$. These deformations limit to finite rotations on the degenerate leaf corresponding to $-{\rm Id}$. boundary holonomy. Standard KAM techniques establish that the action is not ergodic on the leaves in a neighborhood of this degenerate leaf.

Within the class of S-unimodal maps with fixed critical order it is shown that the existence of an absolutely continuous invariant probability measure is not a topological invariant.

We study some diffeomorphisms in the boundary of the set of Anosov diffeomorphisms mainly from the ergodic viewpoint. We prove that these diffeomorphisms, obtained by isotopy from an Anosov $f:M \mapsto M$ through a heteroclinic tangency, determine a manifold ${\cal M}$ of finite codimension in the set of $C^r$ diffeomorphisms. We prove that any diffeomorphism $F$ in ${\cal M}$ is conjugate to $f$; moreover, there exists a unique SRB measure for $F$, and $F$ is Bernoulli with respect to this measure. In particular, if the dimension of $M$ is two, and $\mu $ is a volume element, we prove that the isotopy can be taken such that the measure is preserved.

Let $\{w_i,p_i\}$ be a contractive iterated function system (IFS) [1, pp. 79–80] with probabilities, i.e. a set of contraction maps $w_i:X\to X$ with associated probabilities $p_i$, $i=1,2,\ldots,N$. We provide a simple proof that for almost every address sequence $\sigma$ and for all $x$ the limit $\lim_{n\to \infty}1/n\sum_{i\le n}f(w_{\sigma_n}\circ w_{\sigma_{n-1}}\circ\cdots\circ w_{\sigma_1}(x))$ exists and is equal to $\int_Xf(z)\,d\mu(z)$, where $\mu$ is the invariant measure of the IFS. This is the so called ‘ergodic property’ for the IFS and was proved by Elton in [3]. However, the uniqueness of the invariant measure was not previously exploited. This provides considerable simplification to the proof.

We give a formula for the sum of the Lyapunov exponents of a nondegenerate polynomial map $f$ of ${\bf C}^{2}$ close to $(z,w)\to (z^d,w^d)$. The formula only involves the behavior of $f$ at infinity. In particular, it follows that the sum only depends on the homogeneous part of $f$ of degree $d$.

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

In 1984, Gromov (see [4] and [6]) introduced the idea of subdividing a ‘branching’ polyhedron into smaller cells and replacing these cells by more complex objects, reminiscent of the growth of multicellular organisms in biology. The simplest situation of this kind is a graph substitution which replaces certain subgraphs in a graph $G$ by bigger finite graphs. The most basic graph substitution is a vertex replacement rule which replaces certain vertices of $G$ with finite graphs. This paper develops a framework for studying vertex replacements and discusses the asymptotic behavior of iterated vertex replacements, the limit objects, and the induced dynamics on the space of infinite graphs from the viewpoint of geometry and dynamical systems.

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

The Collet–Eckmann condition is quasisymmetrically invariant: if an S-unimodal map $f$ is quasisymmetrically conjugate to an S-unimodal map $g$ then $f$ satisfies the Collet–Eckmann condition if and only if $g$ satisfies the Collet–Eckmann condition.

Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and a function $F(x,y)$ of period one in both $x$ and $y$ such that $(X,Y)=(a-F_y,b+F_x)$. For a given $F$, let $P$ be the map sending $(x,y)$ into $(F_y(x,y),-F_x(x,y))$. Let $A$ be the image of the torus under this map and let $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive if and only if $a/b$ is irrational. For any point in $A$ but not in $B$ the flow is not metrically transitive. Moreover, if $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.

According to a theorem of Moser, every monotone twist map $\varphi$ on the cylinder ${\Bbb S}^1\times {\Bbb R}$, which is integrable outside a compact set, is the time-1-map $\varphi_H^1$ of a fibrewise convex Hamiltonian $H$. In this paper we prove that if this particular flow $\varphi_H^t$ is also integrable outside a compact set, then $\varphi$ has to be integrable on the whole cylinder (and vice versa, of course). From this dynamical point of view, integrable twist maps appear to be quite rigid.

As is shown in the appendix, an analogous rigidity result becomes trivial in higher dimensions.

Let $ \Gamma $ be a periodic graph with the vertex set ${\Bbb Z}^d $. A subgraph of $ \Gamma $ is called an essential spanning forest if it contains all vertices of $ \Gamma $, has no cycles, and if all its connected components are infinite. The set of all essential spanning forests in $ \Gamma $ is compact in a suitable topology, and ${\Bbb Z}^d $ acts on it by translations. Burton and Pemantle computed the topological entropy of such an action. Their formula turned out to be the same as the formula for the topological entropy of ${\Bbb Z}^d $-actions on certain subgroups of $({\Bbb R}/{\Bbb Z})^{{\Bbb Z}^d}$ obtained previously by Lind, Schmidt and Ward. The question was to explain the coincidence. Here we prove directly that the entropies for two systems must be equal.

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.

Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.

In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.

In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.

By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.