The existence of heteroclinic and homoclinic solutions which shadow corresponding chains of such solutions is established for a class of reversible Hamiltonian systems. The proof involves elementary minimization arguments.

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

This paper studies a special system of $N$-coupled oscillators under white noise. It is shown that we can get a positive recurrent diffusion process by winding the solution of this system on a cylinder along a special direction. Thus defining the rotation number as the limit of the frequency of the oscillator, we can prove that the rotation number exists and the frequency locking is successful no matter how large the white noise is.

The function $d$ defined in (0.3) may not be a metric unless \begin{equation} \phi_p(a_1,\dots, a_p)\le \phi_{p+1} (a_1,\dots, a_{p+1}),\quad \text{for all } (a_i)^{p+1}_1 \in \Gamma^{p+1},\ p=1,\dots.\tag{A} \end{equation} Let $$ \tilde\phi_p(a_1,\dots, a_p)=\max_{1\le i\le p}\phi_1 (a_1,\dots, a_i),\quad (a_i)^p_1\in\Gamma^p,\ p=1,\dots. $$ Then $\tilde\phi$ satisfies (A) and (0.1). Note that \begin{equation} \phi_p(a_1,\dots, a_p) \le\tilde\phi_p (a_1,\dots, a_p)\le p \max_{1\le i\le n} \phi_1(a_i).\tag{B} \end{equation} In (0.3) replace the equality $d(a,b)=e^{-\phi_p(a_1,\dots,a_p)}$ by $d(a,b)=e^{-\tilde\phi_p(a_1,\dots,a_p)}$. Then $d(a,b)\le \max(d(a,c), d(c,b))$. Let \begin{gather*} \tilde\psi_p(x):=\max_{1\le i\le p} \psi_i(x),\ x\in\Gamma^\infty,\quad \tilde\alpha_p(\mu) :=\int \tilde\psi_p\,d\mu,\ p=1, \dots\\ \tilde\alpha(\mu) :=\lim_{p\rightarrow\infty} \frac{\tilde\alpha_p(\mu)}{p}. \end{gather*} Then (B) yields $$ \lim_{p\rightarrow\infty} \frac{\psi_p(x)}{p} = s\Rightarrow \lim_{p\rightarrow\infty} \frac{\tilde\psi_p(x)}{p}=s. $$

Iosifescu pointed out that earlier attempts to generalize Kuzmin's theorem to fibred systems have a gap. In this paper a new proof is given. The convergence rate is worse, but the result applies to multi-dimensional continued fractions and complex continued fractions.

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

Cet article est consacré à la théorie ergodique du flot géodésique sur le fibré unitaire tangent d'une variété hyperbolique relativement à la mesure de Sullivan associée à une mesure conforme de Patterson–Sullivan. Nous prouvons notamment que si le flot géodésique est ergodique alors il est rationnellement ergodique de type asymptotique donné par la série de Poincaré du groupe d'isométries discret associé. Pour ce faire, de nouveaux résultats sur les mesures de Patterson–Sullivan sont introduits. Nous obtenons en conséquence des résultats généraux d'équirépartition asymptotique concernant la série de Poincaré. Nous approfondissons sur d'autres points les propriétés ergodiques du flot géodésique. Ces résultats se généralisent directement pour un groupe agissant sur un espace CAT$(-1)$.

It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases we show that they may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property.

For an inclusion $F < G < L$ of connected real algebraic groups such that $F$ is epimorphic in $G$, we show that any closed $F$-invariant subset of $L/\Lambda$ is $G$-invariant, where $\Lambda$ is a lattice in $L$. This is a topological analogue of a result due to S. Mozes, that any finite $F$-invariant measure on $L/\Lambda$ is $G$-invariant.

This result is established by proving the following result. If in addition $G$ is generated by unipotent elements, then there exists $a\in F$ such that the following holds. Let $U\subset F$ be the subgroup generated by all unipotent elements of $F$, $x\in L/\Lambda$, and $\lambda$ and $\mu$ denote the Haar probability measures on the homogeneous spaces $\overline{Ux}$ and $\overline{Gx}$, respectively (cf. Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as $n\to\infty$.

We also give an algebraic characterization of algebraic subgroups $F<{\rm SL}_n(\mathbb{R})$ for which all orbit closures on ${\rm SL}_n(\mathbb{R})/{\rm SL}_n(\Z)$ are finite-volume almost homogeneous, namely the smallest observable subgroup of ${\rm SL}_n(\mathbb{R})$ containing $F$ should have no non-trivial algebraic characters defined over $\mathbb{R}$.

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

Consider a nice hyperbolic dynamical system (singularities not excluded). Statements about the topological smallness of the subset of orbits, which avoid an open subset of the phase space (for every moment of time, or just for a not too small subset of times), play a key role in showing hyperbolicity or ergodicity of semi-dispersive billiards, especially, of hard-ball systems. As well as surveying the characteristic results, called ball-avoiding theorems, and giving an idea of the methods of their proofs, their applications are also illustrated. Furthermore, we also discuss analogous questions (which had arisen, for instance, in number theory), when the Hausdorff dimension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball-avoiding subsets are naturally related to repellers extensively studied by physicists. For the interested reader we also sketch some analytical and rigorous results about repellers and escape times.

For $1\le r<\infty$, we construct a piecewise $C^{r}$ expanding map $F:D\to D$ on the domain $D=(0,1)\times (-1,1)\subset\mathbb{R}^{2}$ with the following property: there exists an open set $B$ in $D$ such that the diameter of $F^{n}(B)$ converges to $0$ as $n\to\infty$ and the empirical measure $n^{-1}\sum_{k=0}^{n-1}\delta_{F^{k}(x)}$ converges to the point measure $\delta_{p}$ at $p=(0,0)$ as $n \to\infty$ for any point $x\in B$.

A conjecture is formulated and discussed which provides a necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup ${\cal G}$ of the orthogonal group $SO(d)$ ($d$ being the dimension of the billiard in question) be transitive on the unit sphere $S^{d-1}$. If $C_1, \dots, C_k$ are the cylindric scatterers of the billiard, then ${\cal G}$ is generated by the embedded Lie subgroups ${\cal G}_i$ of $SO(d)$, where ${\cal G}_i$ consists of all transformations $g\in SO(d)$ of ${\Bbb R}^d$ that leave the points of the generator subspace of $C_i$ fixed ($1 \le i \le k$). In this paper we can prove the necessity of our conjecture and we also formulate some notions related to transitivity. For hard ball systems, we can also show that the transitivity holds in general: for an arbitrary number $N\ge 2$ of balls, arbitrary masses $m_1, \dots, m_N$ and in arbitrary dimension $\nu \ge 2$. This result implies that our conjecture is stronger than the Boltzmann–Sinai ergodic hypothesis for hard ball systems. We also note a somewhat surprising characterization of the positive subspace of the second fundamental form for the evolution of a special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of sufficiency of an orbit segment.

A class of transformations on $[0,1]^2$, which includes transformations obtained by a Poincaré section of the Lorenz equation, is considered. We prove a formula which connects local dimension, entropy and characteristic exponents of ergodic invariant probability measures.

We give a simple proof for monotonicity of topological entropy in the quadratic family. We use a spectral property of a Ruelle operator.

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$.

In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve.

In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.