Assuming that the underlying probability space is non-atomic, we prove that products of random matrices (linear cocycles) with simple Lyapunov spectrum form an $L^p$-dense set ($1 \leq p < \infty$) in the space of all cocycles satisfying the integrability conditions of the multiplicative ergodic theorem. However, the linear cocycles with one-point spectrum are also $L^p$-dense. Further, in any $L^\infty$-neighborhood of an orthogonal cocycle there is a diagonalizable cocycle.

For products of independent identically distributed random matrices (with distribution $\mu$), simplicity of the Lyapunov spectrum holds on a set of $\mu$'s which is open and dense in both the topology of total variation and the topology of weak convergence, hence is generic in both topologies. For products of matrices which form a Markov chain, the spectrum is simple on a set of transition functions dense in the topology of weak convergence.

Central Cantor sets form a class of symmetric Cantor sets of the real line. Here we give a complete characterization of the $C^{k + \alpha}$ regularity of these Cantor sets. We also give a classification of central Cantor sets up to global and local diffeomorphisms. Examples of central Cantor sets with special dynamical and measure-theoretical properties are also provided. Finally, we calculate the fractal dimensions of an arbitrary central Cantor set.

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed in [7] that \[ \lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq = h_{\rm top}, \] where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of $M$.

In this paper we exhibit an open set of metrics on the two-sphere for which \[ \limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top}, \] for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé in [7].

Let $f$ be a flip-invariant Hölder continuous function on the unit tangent bundle $T^1 M$ of a closed negatively curved Riemannian manifold $M$. We show that conditionals on strong unstable manifolds of the Gibbs equilibrium state defined by $f$ can be realized as Hausdorff measures. Moreover, cohomology classes of flip invariant cocycles are in one-to-one correspondence to cross ratios on the space of four pairwise distinct points of the ideal boundary of the universal covering $\tilde M$ of $M$.

In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equivalence. This is shown to be an equivalence relation. Associated to each size is an entropy which is an equivalence invariant. As in the one-dimensional case this entropy is either the classical entropy or is zero. Several examples are discussed.

In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

We study families $G(\lambda,\cdot)$ of entire functions that are approximated by a sequence of families $G_n(\lambda,\cdot)$ of entire functions, where $\lambda\in\C$ is a parameter. In order to control the dynamics, the families are assumed to be of the same constant finite type. In this setting we prove the convergence of the hyperbolic components in parameter space as kernels in the sense of Carathéodory.

In this paper we study a generalization of Mayer's result on the Selberg zeta function of $PSL(2, {\Bbb Z})$. Let $\Gamma$ be a cofinite Fuchsian group. We construct a Markov system ${\cal T}_{\Gamma}$ by modifying the Bowen–Series construction of a Markov map $T_{\Gamma}$ associated with $\Gamma$. The Markov system enables us to define transfer operators $L(s)$ for ${\cal T}_{\Gamma}$ so that they determine a meromorphic function taking values with nuclear operators on a nice function space. We show that the Selberg zeta function $Z(s)$ of $\Gamma$ has a determinant representation $Z(s)=\Det(I-L(s))F(s)$, where $\Det(I-L(s))$ is the Fredholm determinant of $L(s)$ and $F(s)$ is a meromorphic function depending only on a finite number of hyperbolic conjugacy classes of $\Gamma$. Combining such a representation and the investigation of the spectral properties of $L(s)$, we can also obtain some analytic information of $Z(s)$.

An interval map with only one discontinuity is isomorphic to a rotation of the circle, and has continuous eigenfunctions. What we show here is that for almost every choice of lengths of the intervals, this is the only way an irreducible interval exchange can have a somewhere continuous eigenfunction. We show slightly more, considering certain towers over the interval exchange, showing that outside of a set of choices for interval lengths of measure zero these have a somewhere continuous eigenfunction only if they are isomorphic to either a rotation, or a tower of constant height over an interval exchange.

Let $ \Phi $ be a $C^2$ codimension one Anosov flow on a compact Riemannian manifold $M$ of dimension greater than three. Verjovsky conjectured that $ \Phi $ admits a global cross-section and we affirm this conjecture when $ \Phi $ is volume preserving in the following two cases: (1) if the sum of the strong stable and strong unstable bundle of $\Phi$ is $ \theta $-Hölder continuous for all $ \theta < 1 $; (2) if the center stable bundle of $ \Phi $ is of class $ C^{1 + \theta} $ for all $ \theta < 1 $. We also show how certain transitive Anosov flows (those whose center stable bundle is $C^1$ and transversely orientable) can be ‘synchronized’, that is, reparametrized so that the strong unstable determinant of the time $t$ map (for all $t$) of the synchronized flow is identically equal to $ e^t $. Several applications of this method are given, including vanishing of the Godbillon–Vey class of the center stable foliation of a codimension one Anosov flow (when $ \dim M > 3 $ and that foliation is $ C^{1 + \theta} $ for all $ \theta < 1 $), and a positive answer to a higher-dimensional analog to Problem 10.4 posed by Hurder and Katok in [HK].

The defect of a factor map out of a totally disconnected dynamical system is a number in $[0,\infty]$ which gives a quantitative measure of how well the dynamical behaviour of the factor agrees with that of the extension. The larger the defect, the more they differ. In this paper we obtain the basic general properties of the defect and calculate it in a series of specific cases.