We prove that the time-one map of the geodesic flow for a closed, negatively curved surface is stably ergodic.

We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function $f$ to be connected or to be disconnected as a subset of the complex plane ${\Bbb C}$. The other is to investigate the boundary of an unbounded periodic Fatou component $U$, which is known to be simply-connected. These are related as follows: let $\varphi : {\Bbb D} \longrightarrow U$ be a Riemann map of $U$ from a unit disk ${\Bbb D}$, then under some mild conditions we show that the set $\Theta_{\infty}$ of all angles where $\varphi$ admits the radial limit $\infty$ is dense in $\partial {\Bbb D}$ if $U$ is an attracting basin, a parabolic basin or a Siegel disk. If $U$ is a Baker domain on which $f$ is not univalent, then $\Theta_{\infty}$ is dense in $\partial {\Bbb D}$ or at least its closure $\overline{\Theta_{\infty}}$ contains a certain perfect set, which means the boundary $\partial U$ has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set $J_f$ in ${\Bbb C}$. If $U$ is a Baker domain on which $f$ is univalent, however, we shall show by giving an example that $\partial U$ can be a Jordan arc in ${\Bbb C}$, which has a rather simple structure, and, moreover, $J_f$ can be connected.

We also consider the connectivity of the set $J_f \cup \{ \infty \}$ in the Riemann sphere $\widehat{{\Bbb C}}$ and show that $J_f \cup \{ \infty \}$ is connected if and only if $f$ has no multiply-connected wandering domains.

In [10] we considered a class of hyperbolic endomorphisms and asked the question whether there exists a physically motivated invariant measure (SRB measure) and if so we gave a criterion when the map is invertible on a set of full measure. In this work we want to consider a particular example of this class — in fact a three-parameter family of those — and prove that a.s. the criterion is fulfilled. From this it follows that the Young formulae for the Hausdorff dimension of the SRB measure holds.

Given a real number $\beta > 1$, we construct a simple purely infinite $C^*$-algebra ${\cal O}_{\beta}$ as a $C^*$-algebra arising from the $\beta$-subshift in the symbolic dynamics. The $C^*$-algebras $\{{\cal O}_{\beta} \}_{1<\beta \in {\Bbb R}}$ interpolate between the Cuntz algebras $\{{\cal O}_n\}_{1 < n \in {\Bbb N}}$. The K-groups for the $C^*$-algebras ${\cal O}_{\beta}$, $1 < \beta \in {\Bbb R}$, are computed so that they are completely classified up to isomorphism. We prove that the KMS-state for the gauge action on ${\cal O}_{\beta}$ is unique at the inverse temperature $\log \beta$, which is the topological entropy for the $\beta$-shift. Moreover, ${\cal O}_{\beta}$ is realized to be a universal $C^*$-algebra generated by $n-1=[\beta]$ isometries and one partial isometry with mutually orthogonal ranges and a certain relation coming from the sequence of $\beta$-expansion of $1$.

In this paper we derive some properties of a variety of entropy that measures rotational complexity of annulus homeomorphisms, called asymptotic or rotational entropy. In previous work [KS] the authors showed that positive asymptotic entropy implies the existence of infinitely many periodic orbits corresponding to an interval of rotation numbers. In our main result, we show that a Hölder $C^1$ diffeomorphism with nonvanishing asymptotic entropy is isotopic rel a finite set to a pseudo-Anosov map. We also prove that the closure of the set of recurrent points supports positive asymptotic entropy for a ($C^0$) homeomorphism with nonzero asymptotic entropy.

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.

We study the box geometry of bi-modal maps. As has been expected, unlike in the unimodal case, there may be non-renormalizable mappings which do not show the decay of geometry. We provide an example. On the other hand, we come up with a natural pattern in which the critical orbits are closely intertwined, but the decay of geometry persists. All this is done in the context of a generally defined box inducing construction for bi-modal maps.

In 1971, M. M. Peixoto [15] introduced an important topological invariant of Morse–Smale flows on surfaces, which he called a distinguished graph $X^*$ associated with a given flow. Here we show how the Peixoto invariant can be essentially simplified and revised by adopting a purely topological point of view connected with the embeddings of arbitrary graphs into compact surfaces. The newly obtained invariant, $X^R$, is a rotation of a graph $X$ generated by a Morse–Smale flow. (A rotation $R$ is a cyclic order of edges given in every vertex of $X$.) The invariant $X^R$ ‘reads-off’ the topological information carried by a flow, being in a one-to-one correspondence with the topological equivalence classes of Morse–Smale flowsAnd foliations, see [3]. We do not treat the case of foliations, bearing in mind that they are defined by involutive flows on covering manifolds [9].. As a counterpart to the equivalence result we prove a realization theorem for an ‘abstractly given’ $X^R$. (Our methods are completely different from those of Peixoto and they clarify the connections between graphs and flows on surfaces.) The idea of ‘rotation systems’ on graphs can be further exploited in the study of recurrent flows (and foliations) with several disjoint quasiminimal sets on surfaces [10].

Le mélange faible topologique de presque tout échange d'intervalles différent d'une rotation a été démontré, en 1990, par Nogueira et Rudolph. On généralise ici ce résultat en considérant des flots suspensions au-dessus d'échanges d'intervalles : on montre que pour presque toute forme différentielle quadratique orientable sur une surface de genre ${\ge}2$, le flot vertical associé est topologiquement faiblement mélangeant ; le cas du tore permettant, lui, d'obtenir des fonctions propres continues naturelles par projection sur l'une des coordonnées.

It is shown that systems with hyperbolic structure have the Bernoulli property. Some new results on smooth cross-sections of hyperbolic Bernoulli flows are also derived. The proofs involve an abstract version of our original methods for showing that the geodesic flow on surfaces of negative curvature are Bernoulli.

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.

We study cohomology-free (c.f.) diffeomorphisms of the torus $T^n$. A diffeomorphism is c.f. if every smooth function $f$ on $T^n$ is cohomologous to a constant $f_0$, i.e. there exists a $C^{\infty}$ function $h$ so that $h-h\circ\varphi=f-f_0$. We show that the only c.f. diffeomorphisms of $T^n$, $1\le n\le3$, are the smooth conjugations of Diophantine translations. For $n=4$, we prove the same result for c.f. orientation-preserving diffeomorphisms.

We apply Thurston's characterization of postcritically finite rational maps as branched coverings of the sphere to give new classes of combination theorems for postcritically finite rational maps. Our constructions increase the degree of the map but always yield branched coverings which are equivalent to rational maps, independent of the combinatorics of the original map. The main tool is a general theorem based on the intersection number of arcs and curves which controls the region in the sphere in which an obstruction may reside.

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 study the local connectedness of the Julia set of the family $f_b(z)=z^m(z^n-b)$. Assuming that the Julia set is connected and depending on whether $m=1$ or $m>1$, a sufficient condition for the local connectivity of the Julia set is obtained. This complements the known results of Yoccoz–Hubbard–Branner for $m=n=1$.

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.

A class of negative definite kernels is defined in terms of measure spaces. Using this concept, property (T) for a countable group $\Gamma$ is characterized in terms of measure-preserving actions of $\Gamma$, as follows. If a set $S$ is translated a finite amount by any fixed element of $\Gamma$, then there is a uniform bound on how far $S$ is translated.

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.

We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.

In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.

Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.