We prove that a polynomial vector field on $\mathbb{C}^n$, $n\geq 2$, whose corresponding projective foliation has only singularities of hyperbolic type is linear in some affine chart provided that it is transverse to a sequence of spheres bounding balls exhausting ${\mathbb{C}}^n$.

Let $f$ be a transcendental meromorphic function and $U$ a Baker domain of $f$. We obtain new estimates for the behaviour of the iterates of $f$ in $U$ and we use these estimates to improve an earlier result relating the geometric properties of $U$ to the proximity of $f$ to the identity function in $U$. We also apply these estimates to Baker domains of transcendental meromorphic functions $f$ of the form

\begin{gather*} f(z) = az + bz^ke^{-z}(1+o(1)) \quad \text{as } \Re (z) \rightarrow \infty, \end{gather*}

where $k \in {\mathbb N},\ a > 1$ and $b > 0$, and show that these Baker domains contain an unbounded set of critical points and an unbounded set of critical values.

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 $n>4$.

If a closed manifold M possesses two Riemannian metrics which have the same unparameterized geodesics and are not strictly proportional at each point, then the topological entropy of both geodesic flows is zero. This is the main result of the paper and it has many dynamical and topological corollaries. In particular, such a manifold M should be finitely covered by the product of a rationally elliptic manifold and a torus.

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital $C^*$-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let $A$ be a stably finite simple unital $C^*$-algebra, and let $\alpha$ be an automorphism of $A$ which has the tracial Rokhlin property. Suppose $A$ has real rank zero and stable rank one, and suppose that the order on projections over $A$ is determined by traces. Then the crossed product algebra $C^* (\mathbb{Z}, A, \alpha)$ also has these three properties. We also present examples of $C^*$-algebras $A$ with automorphisms $\alpha$ which satisfy the above assumptions, but such that $C^* (\mathbb{Z}, A, \alpha)$ does not have tracial rank zero.

We generalize the notion of reversible systems in symbolic dynamics, and investigate their properties. It is shown that a reversible topological Markov shift can be represented by a pair of matrices of special types. This enables us to classify the invariant measures of the reversible systems. Necessary and/or sufficient conditions for the existence of a reversal of finite order are established in terms of the adjacency matrices. We also prove that a topological Markov shift with a reversal of order two admits reversals of all orders.

This is a study of the invariant measures of a synchronized subshift (in particular, a sofic shift). Among other results we give necessary and sufficient conditions for the existence of a maximal measure of full support, and a method to determine if the shift is intrinsically ergodic.

Measures associated to (compact, non-singular) hyperbolic surface laminations are studied. The main results of this work relate harmonic measures and measures invariant under the laminated geodesic and horocycle flows.

Let $f:M\to M$ be a partially hyperbolic diffeomorphism, ${\it TM}=E^{ss}\oplus E^c\oplus E^{uu}$ such that the stable foliation $\mathcal{F}^{ss}(f)$ is minimal. We give a sufficient condition so that this foliation remains minimal after perturbations, i.e. $\mathcal{F}^{ss}(g)$ is minimal for every g sufficiently close to f.

We introduce a general class of automorphisms of rotation algebras, the non-commutative Furstenberg transformations. We prove that fully irrational non-commutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar's Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the $C$*-algebras of discrete subgroups of five-dimensional nilpotent Lie groups, considered by Milnes and Walters, are crossed products of simple $C$*-algebras ($C$*-algebras of minimal ordinary Furstenberg transformations) by automorphisms which have the tracial Rokhlin property. It follows that these algebras also have stable rank one, real rank zero, and order on projections determined by traces.

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275). More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally) Lipschitz functions from $B$ to $\mathbb{R}^{m}$ that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function $f \in M$, the Hausdorff dimension of $f(X)$ is at least $\min\{ m, d / (1 + \tau) \}$. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on $X$. The factor $1 / (1 + \tau)$ can be improved to $1 / (1 + \tau / 2)$ if $B$ is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when $\tau = 0$. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case $\tau > 0$.

We show that the volume entropy of surfaces with unit systole is bounded from above by a constant which does not depend on the metric, which answers a question raised by A. Katok. This upper bound follows from a comparison between the geometric and algebraic lengths on the fundamental groups of surfaces.

We obtain a local topological and dynamical description of expansive attractors on surfaces. The main result is that expansive attractors on surfaces are hyperbolic and have local product structure, except possibly at a finite number of periodic points, which can be either sinks, singularities or épines. Some open questions concerning this kind of dynamics are posed.

This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563–603). We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere $\overline{\mathbb C}$ of degree two or greater, the Julia set of any subsemigroup of G is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic, then the fiberwise Julia sets are porous, and the dynamics is weakly rigid. Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise maps are polynomials, then under some conditions, the fiberwise basins of infinity are John domains. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on $\overline{\mathbb C}$) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

Generalizing a three-dimensional result of Doering, we prove that a robustly transitive vector field defined on a compact manifold of any dimension greater than three admits no fixed point.