It is shown that, for the von Neumann algebra $A$ obtained from a principal measured groupoid $\mathcal{R}$ with the diagonal subalgebra $D$ of $A$, there exists a natural ‘bijective’ correspondence between coactions on $A$ that fix $D$ pointwise and Borel 1-cocycles on $\mathcal{R}$. As an application of this result, we classify a certain type of coactions on approximately finite-dimensional type II factors up to cocycle conjugacy. By using our characterization of coactions mentioned above, we are also able to generalize to some extent those results of Zimmer concerning 1-cocycles on ergodic equivalence relations into compact groups.

We establish the existence of center manifolds for a class of non-uniformly partially hyperbolic trajectories of non-autonomous ordinary differential equations $v'=A(t)v+f(t,v)$ on Banach spaces. In particular, we allow the stable and unstable components of $v'=A(t)v$ to exhibit non-uniform contraction and expansion along the trajectory. We also allow the center component to exhibit non-uniform behavior. To the best of our knowledge, we establish the first center manifold theorem in the non-uniform setting.

Ruelle proved that for quasiconformal deformations of cocompact Fuchsian groups, the Hausdorff dimension of the limit set is an analytic function of the deformation. In this paper, we give a criterion for the failure of analyticity for certain infinitely generated groups. In particular, we show that it fails for any infinite abelian cover of a compact surface, answering a question of Astala and Zinsmeister.

We present a generalization of the functional equation for the Weierstrassk $\wp$-function for hyperelliptic surfaces of infinite genus arising from iteration of the horseshoe and baker maps. The ramified cover of these infinite genus surfaces over the complex plane are associated to a quadratic differential of finite norm with simple poles accumulating to infinity. We study the geometry of its critical trajectories emanating from these poles and their rate of accumulation.

We study the dynamics near transverse intersections of stable and unstable manifolds of sheets of symmetric periodic orbits in reversible systems. We prove that the dynamics near such homoclinic and heteroclinic intersections is not $C^1$ structurally stable. This is in marked contrast to the dynamics near transverse intersections in both general and conservative systems, which can be $C^1$ structurally stable. We further show that there are infinitely many sheets of symmetric periodic orbits near the homoclinic or heteroclinic orbits. We establish the robust occurrence of heterodimensional cycles, that is, heteroclinic cycles between hyperbolic periodic orbits of different index, near the transverse intersections. This is shown to imply the existence of hyperbolic horseshoes and infinitely many periodic orbits of different index, all near the transverse intersections.

Let $X$ be a one-sided subshift of finite type on a countable alphabet, and $T:X\to X$ the shift map. If $f:X\to{\mathbb R}$ is continuous, we provide conditions guaranteeing that $f$-maximizing measures exist, and are characterised by the condition that their support lies in a certain compact set.

We investigate the ideal structures of the $C^*$-algebras arising from topological graphs. We give a complete description of ideals of such $C^*$-algebras that are invariant under the so-called gauge action, and give a condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our $C^*$-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph for which the associated $C^*$-algebra is prime but not primitive.

We show that the Voiculescu–Brown entropy of a non-commutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$ is at least half the value of the topological entropy of the corresponding classical toral automorphism. We also obtain some information concerning the positivity of local Voiculescu–Brown entropy with respect to single unitaries. In particular we show that if $S$ has no roots of unity as eigenvalues then the local Voiculescu–Brown entropy with respect to every product of canonical unitaries is positive, and also that in the presence of completely positive CNT entropy the unital version of local Voiculescu–Brown entropy with respect to every non-scalar unitary is positive.

On s'intéresse aux propriétés asymptotiques de la suite des discrétisations uniformes d'un homéomorphisme du cercle. On étudie leurs dépendances en les propriétés arithmétiques du nombre de rotation de l'homéomorphisme en question. En particulier, on considère la structure des cycles périodiques, leurs distributions spatiales et le temps de stabilisation du processus de discrétisation.

We study mixing properties of algebraic actions of $\mathbb Q^d$, showing in particular that prime mixing $\mathbb Q^d$ actions on connected groups are mixing of all orders, as is the case for $\mathbb Z^d$-actions. This is shown using a uniform result on the solution of $S$-unit equations in characteristic zero fields due to Evertse, Schlickewei and W. Schmidt. In contrast, algebraic actions of the much larger group $\mathbb Q^*$ are shown to behave quite differently, with finite order of mixing possible on connected groups.

A framework for understanding the geometry of continuous actions of $\mathbb Z^d$ was developed by Boyle and Lind using the notion of expansive behaviour along lower-dimensional subspaces. For algebraic $\mathbb Z^d$-actions of entropy rank one, the expansive subdynamics are readily described in terms of Lyapunov exponents. Here we show that periodic point counts for elements of an entropy rank-one action determine the expansive subdynamics. Moreover, the finer structure of the non-expansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.

In this paper we revisit once again, see Shub and Sullivan (Ergod. Th. & Dynam. Sys.5 (1985), 285–289), a family of expanding circle endomorphisms. We consider a family $\{B_\theta\}$ of Blaschke products acting on the unit circle, $\mathbb{T}$, in the complex plane obtained by composing a given Blaschke product $B$ with the rotations about zero given by multiplication by $\theta \in \mathbb{T}$. While the initial map $B$ may have a fixed sink on $\mathbb{T}$, there is always an open set of $\theta$ for which $B_\theta$ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of $\int \ln|B'(z)|\,{\it dz}$.

For the family of exponential maps $E_{\kappa}(z)=\exp(z)+\kappa$, we prove an analog of Böttcher's theorem by showing that any two exponential maps $E_{\kappa_1}$ and $E_{\kappa_2}$ are conjugate on suitable subsets of their escaping sets, and this conjugacy is quasiconformal. Furthermore, we prove that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's ‘pinched disk model’ for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping points. We also answer several questions about escaping endpoints of dynamic rays. In particular, we give a necessary and sufficient condition for the ray to be continuously differentiable in such a point, and show that escaping points can escape arbitrarily slowly. Furthermore, we show that the principle of topological renormalization is false for attracting exponential maps.

A $C^\infty$ diffeomorphism $\varphi$ of a manifold $M$ is cohomologically rigid if for each smooth function $f$ on $M$ there is a constant $f_0$ so that the cohomological equation $h-h\circ\varphi=f-f_0$ has a smooth solution $h$. We prove that all of the eigenvalues of the mapping on $H_1(\mathbb{T}^n,\mathbb{ R})$ induced by a cohomologically rigid diffeomorphism $\varphi$ of the torus $\mathbb{T}^n$ are roots of unity if $n<4$. The same is true for $n=4$ provided that $\varphi$ preserves orientation. We do not know whether it is true when $n=4$ and $\varphi$ reverses orientation or when $n>4$.