We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distribution are related to the Riemann zeta function.

Let $f_1$ and $f_2$ be rational maps with Julia sets $J_1$ and $J_2$, and let $\Psi:J_1\to \mathbb{P}^1$ be any continuous map such that $\Psi\circ f_1=f_2\circ \Psi$ on $J_1$. We show that if $\Psi$ is $\mathbb{C}$-differentiable, with non-vanishing derivative, at some repelling periodic point $z_1\in J_1$, then $\Psi$ admits an analytic extension to $\mathbb{P}^1\setminus {\mathcal E}_1$, where ${\mathcal E}_1$ is the exceptional set of $f_1$. Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci. École Norm. Sup. (3) 40 (1923), 97–150). Furthermore, if ${\mathcal E}_1=\emptyset$ then the extended map $\Psi$ is rational, and in this situation $\Psi(J_1)=J_2$ and $\Psi^{-1}(J_2)=J_1$, provided that $\Psi$ is not constant. On the other hand, if ${\mathcal E}_1\neq \emptyset$ then the extended map may be transcendental: for example, when $f_1$ is a power map (conjugate to $z\mapsto z^{\pm d}$) or a Chebyshev map (conjugate to $\pm \text{Х}_d$ with $\text{Х}_d(z+z^{-1}) = z^d+z^{-d}$), and when $f_2$ is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz1(4) (1989), 102–116) proved that these are the only such examples. We present a new proof.

We show that there is a spectral gap for the transfer operator associated with a rational map $f$ on the Riemann sphere. Using this and the method of perturbed operators, we establish the (local) central limit theorem for the measure of maximal entropy of $f$, with an estimate on the speed of convergence.

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a one-dimensional map.

Let $T(1),\dots,T(d)$ be conservative, invertible, non-singular, commuting transformations on the Polish measure space $(X,m)$. Then for $f$ and $p$ in $L^1(m)$ with $p>0$,

\[ \frac{{\hat T}(1)_{-N}^N \dotsb {\hat T}(d)_{-N}^Nf}{{\hat T}(1)_{-N}^N \dotsb {\hat T}(d)_{-N}^Np}\to E[f | {\mathcal I}]/E[p| {\mathcal I}]\quad \text{as }N\to\infty. \]

In this paper some families of skew product self-maps $F$ on the square are considered. The main example is a family forming a two-dimensional analogue of the tent map family. According to the assumptions made in this paper these maps are almost injective. This means that the points of the attractor having more than one inverse image form a set of measure zero for all interesting measures. It may be that $F$ does not have a finite Markov partition. The Hausdorff dimension of the attractor is computed.

We study curves of fixed points for certain diffeomorphisms of ${\mathbb{R}}^3$ that are induced by automorphisms of a trace algebra. We classify these curves. There is a function $E$ which is invariant under all such trace maps and the level surfaces $E_t: E=t$ are invariant; a point of $E_t$ will be said to have level $t$. The surface $E_1$ is significant. Then most fixed points on $E_1$ are actually on a curve $\gamma$ of fixed points interior to $E_1$. We describe the possibilities for the other end of $\gamma$ on $E_1$.

We consider a new orbit equivalence invariant for measure-preserving actions of groups on the probability space, $\sigma:G\rightarrow {\rm Aut}(X,\mu)$, denoted by $\chi_0(\sigma;G)$ and defined as the ‘intersection’ of the 1-cohomology group, H$^1(\sigma,G)$, with Connes' invariant, $\chi(M)$, of the cross product von Neumann algebra, $M=L^\infty(X,\mu)\rtimes_\sigma G$. We calculate $\chi_0(\sigma;G)$ for certain actions of groups of the form $G=H\times K$ with $H$ non-amenable and $K$ infinite amenable and we deduce that any such group has uncountably many orbit inequivalent actions.

One of the fundamental properties of the Mandelbrot set is that the set of postcritically finite parameters is structured like a tree. We extend this result to the set of quadratic kneading sequences and show that this space contains no irrational decorations. Along the way, we prove a combinatorial analogue to the correspondence principle of dynamic and parameter rays. Our key tool is to work simultaneously with the two equivalent combinatorial concepts of Hubbard trees and kneading sequences.

It is known that the closure ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ of the orbit ${\mathop{\hbox{\rm Orb}}\nolimits}_{g}(x)$ of a point $x$ of a compact nilmanifold $X$ under a polynomial sequence $g$ of translations of $X$ is a disjoint finite union of sub-nilmanifolds of $X$. Assume that $g(0)=1_{G}$ and let $A$ be the group generated by the elements of $g$; we show in this paper that for almost all points $x\in X$, the closures ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ are congruent (that is, are translates of each other), with connected components of ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ equal to (some of) the connected components of ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{A}(x)$.

We prove the existence of $C^{1}$ critical subsolutions of the Hamilton–Jacobi equation for a time-periodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system.

We prove that the Collet–Eckmann condition for recurrent critical orbits inside the Julia set of a rational map with no parabolic periodic orbits implies uniform hyperbolicity on periodic orbits.

Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.

We study a new space, $R(X)$, of real-valued continuous functions on the space $X$ of sequences of zeros and ones. We show exactly when the Ruelle operator theorem holds for such functions. Any $g$-function in $R(X)$ has a unique $g$-measure and powers of the corresponding transfer operator converge. We also show Bow$(X,T)\neq W(X,T)$ and relate this to the existence of bounded measurable coboundaries, which are not continuous coboundaries, for the shift on the space of bi-sequences of zeros and ones.

First notions of relative complexity function and relative sensitivity are introduced. It turns out that for any open factor map $\pi: (X, T)\rightarrow (Y, S)$ between topological dynamical systems with minimal $(Y, S),\ \pi$ is positively equicontinuous if and only if the relative complexity function is bounded for each open cover of $X$; and that any non-trivial weakly mixing extension is relatively sensitive. Moreover, a relative version of the notable result that any $M$-system is sensitive if it is not minimal is obtained. Then notions of relative scattering and relative Mycielski's chaos are introduced. A relative disjointness theorem involving relative scattering is given. A relative version of the well-known result that any non-trivial scattering topological dynamical system is Li–Yorke chaotic is proved.