We study the existence of models with minimal topological entropy among the class of all continuous maps from a given graph to itself with a fixed behavior on a given finite invariant set.

We construct a transcendental entire function $f$ with $J(f)=\mathbb{C}$ such that $f$ has arbitrarily slow growth; that is, $\log |f(z)|\leq\phi(|z|)\log |z|$ for $|z|>r_0$, where $\phi$ is an arbitrary prescribed function tending to infinity.

The resonant tori of an integrable system are destroyed by a perturbation. If the Hamiltonian is convex, they give rise to hyperbolic lower-dimensional invariant tori or to Aubry–Mather invariant sets. Bolotin has proved the existence of homoclinic orbits to the hyperbolic tori but not to the Aubry–Mather invariant sets. We solve this problem and obtain, for each resonant frequency, the existence of an invariant set with homoclinic orbits.

Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.

We study the motion of a charge on a conformally flat Riemannian torus in the presence of a magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ‘twisted’ version of the recent generalization of Hopf's rigidity result obtained by Burago and Ivanov.

We study smooth local models of families of symmetric and reversible vector fields near a partially hyperbolic singularity. Special attention is given to the question of whether the involved changes of variables commute with the symmetry.

We first study situations where the stable AF algebras defined by two square primitive non-singular incidence matrices with non-negative integer matrix elements are isomorphic, even though no powers of the associated automorphisms of thecorresponding dimension groups are isomorphic. More generally we consider necessary and sufficient conditions for two such matrices to determine isomorphic dimension groups.We give several examples.

Consider a finite Abelian group $(G,+)$, with $|G|=p^r$, $p$ a prime number, and $\varphi: G^\mathbb{N} \to G^\mathbb{N}$ the cellular automaton given by $(\varphi x)_n=\mu x_n+\nu x_{n+1}$ for any $n\in \mathbb{N}$, where $\mu$ and $\nu$ are integers coprime to $p$. We prove that if $\mathbb{P}$ is a translation invariant probability measure on $G^\mathbb{Z}$ determining a chain with complete connections and summable decay of correlations, then for any ${\underline w}= (w_i:i<0)$the Cesàro mean distribution $${\cal M}_{\mathbb{P}_{\underline w}} =\lim_{M\to\infty} \frac{1}{M} \sum^{M-1}_{m=0}\mathbb{P}_{\underline w}\circ\varphi^{-m},$$ where $\mathbb{P}_{\underline w}$ is the measure induced by $\mathbb{P}$ on $G^\mathbb{N}$ conditioned by $\underline w$, exists and satisfies ${\cal M}_{\mathbb{P}_{\underline w}}=\lambda^\mathbb{N}$, the uniform product measure on $G^\mathbb{N}$. The proof uses a regeneration representation of $\mathbb{P}$.

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is $2(n-1)$. In the case when the perturbation is quadratic ($n=2$) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.

Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.

Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.

The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

We describe a strategy which allows one to produce computer assisted proofs of almost everywhere strong convergence of Jacobi–Perron type algorithms in arbitrary dimension. Numerical work is carried out in dimension three to illustrate our method. To the best of our knowledge this is the first result on almost everywhere strong convergence in dimension greater than two.

We study the link between variations of entropy on a compact hyperbolic manifold $M$ and infinitesimal flat conformal deformations of $M$. We remark that the entropy of the Liouville measure increases in the direction of these deformations. We then explain a new construction of infinitesimal flat conformal deformations by bending along totally geodesic hypersurfaces of $M$ which allows us to extend a theorem of L. Flaminio.

We show that if two aperiodic automorphisms of a separable nuclear unital purely infinite simple $C^{\ast}$-algebra are asymptotically unitarily equivalent, then they are outer conjugate with respect to an automorphism which is isotopic to the identity automorphism. Thus, by Kirchberg and Phillips, they have the same KK-class if and only if they are outer conjugate with respect to an automorphism which is in the KK-class of the identity automorphism.

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.

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$(-1)$.

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 $x\in B$.

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.