In this paper we study certain conformal iterated function schemes in two dimensions that are natural generalizations of the Sierpinski carpet construction. In particular, we consider scaling factors for which the open set condition fails. For such ‘fat Sierpinski carpets’ we study the range of parameters for which the dimension of the set is exactly known, or for which the set has positive measure.

We investigate the ideal structures of the $C^*$-algebras arising from topological graphs. We give a complete description of ideals of such $C^*$-algebras that are invariant under the so-called gauge action, and give a condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our $C^*$-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph for which the associated $C^*$-algebra is prime but not primitive.

We construct a homeomorphism from the filled Julia set of a given polynomial into the connectedness locus of the family of all polynomials of higher degree by using intertwining surgery. A polynomial in the image of such a homeomorphism has a renormalization of ‘capture’ type and the inverse map is given by the ‘marking’ of the captured critical orbit.

We consider locally compact groups $G$ admitting a topologically transitive $\mathbb{Z}^d$-action by automorphisms. It is shown that such a group $G$ has a compact normal subgroup $K$ of $G$, invariant under the action, such that $G/K$ is a product of (finitely many) locally compact fields of characteristic zero; moreover, the totally disconnected fields in the decomposition can be chosen to be invariant under the $\mathbb{Z}^d$-action and such that the $\mathbb{Z}^d$-action is via scalar multiplication by non-zero elements of the field. Under the additional conditions that $G$ be finite dimensional and ‘locally finitely generated’ we conclude that $K$ as above is connected and contained in the center of $G$. We describe some examples to point out the significance of the conditions involved.

We recall the problem posed by Peres and Solomyak in Problems on self-similar and self-affine sets; an update. Progr. Prob.46 (2000), 95–106: can one find examples of self-similar sets with positive Lebesgue measure, but with no interior? The method in Properties of measures supported on fat Sierpinski carpets, this issue, leads to families of examples of such sets.

We show that the Voiculescu–Brown entropy of a non-commutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$ is at least half the value of the topological entropy of the corresponding classical toral automorphism. We also obtain some information concerning the positivity of local Voiculescu–Brown entropy with respect to single unitaries. In particular we show that if $S$ has no roots of unity as eigenvalues then the local Voiculescu–Brown entropy with respect to every product of canonical unitaries is positive, and also that in the presence of completely positive CNT entropy the unital version of local Voiculescu–Brown entropy with respect to every non-scalar unitary is positive.

We carry the argument used in the proof of Denjoy's theorem over to the quasiperiodically forced case. Thus we derive that, if a system of quasiperiodically forced circle diffeomorphisms with bounded variation of the derivative has no invariant graphs with a certain kind of topological regularity, then the system is topologically transitive.

There is a natural action of SL$(2,\mathbb{R})$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \big\{\big(\begin{smallmatrix}1 & * \\ 0 & 1\end{smallmatrix}\big)\big\}$. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL$(2,\mathbb{R})$-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an analogue of Ratner's theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is $2 \pi/n$, with $n \ge 5$ and n odd.

We will show that an equivalence relation on a Cantor set arising from a two-dimensional substitution tiling by polygons is affable in the sense of Giordano, Putnam and Skau.

We prove the equidistribution of Hecke points for any connected non-compact $\mathbb{Q}$-simple real algebraic group G and an arithmetic subgroup $\Gamma\subset G(\mathbb{Q})$, generalizing a theorem of Clozel, Oh and Ullmo. The main tool is a theorem of Mozes and Shah on unipotent flows.

For each $n>1$, we construct explicitly a rigid weakly mixing rank-$n$ transformation with homogeneous spectrum of multiplicity $n$. The existence of such transformations was established recently by O. Ageev via Baire category arguments (a new short category proof is also given here). As an application, for any subset $M\subset\mathbb N$ containing 1, a weakly mixing transformation whose essential range for the spectral multiplicity equals $n\cdot M$ is constructed.

It is well known that in the dynamics of a piecewise strictly monotone (that is, piecewise embedding) map $f$ on an interval, the topological entropy can be expressed in terms of the growth of the number (that is, the lap number) of strictly monotone intervals for $f^n$. Recently, there has been an increase in the importance of fractal sets in the sciences, and many geometric and dynamical properties of fractal sets have been studied. In the present paper, we shall study topological entropy of some maps on regular curves, which are contained in the class of fractal sets. We generalize the theorem of Misiurewicz–Szlenk and Young to the cases of regular curves and dendrites.

On s'intéresse aux propriétés asymptotiques de la suite des discrétisations uniformes d'un homéomorphisme du cercle. On étudie leurs dépendances en les propriétés arithmétiques du nombre de rotation de l'homéomorphisme en question. En particulier, on considère la structure des cycles périodiques, leurs distributions spatiales et le temps de stabilisation du processus de discrétisation.

In this work we construct an example of a smooth dynamical system for which the minimal and statistical attractors do not coincide. In the language of SRB measures, this is an example for which there exists a unique natural measure, but there is no observable one. Also, as the statistical attractor describes the mean behaviour of individual points, and the minimal one the mean behaviour of the Lebesgue measure, this example provides us with a dynamical realization of the Riesz example.

We study mixing properties of algebraic actions of $\mathbb Q^d$, showing in particular that prime mixing $\mathbb Q^d$ actions on connected groups are mixing of all orders, as is the case for $\mathbb Z^d$-actions. This is shown using a uniform result on the solution of $S$-unit equations in characteristic zero fields due to Evertse, Schlickewei and W. Schmidt. In contrast, algebraic actions of the much larger group $\mathbb Q^*$ are shown to behave quite differently, with finite order of mixing possible on connected groups.

In this paper we study the existence of analytic families of reducible linear quasi-periodic differential equations in matrix Lie algebras. Under suitable conditions we show, by means of a Kolmogorov–Arnold–Moser (KAM) scheme, that a real analytic quasi-periodic system close to a constant matrix can be modified by the addition of a time-free matrix that makes it reducible to constant coefficients. If the system depends analytically on external parameters, then this modifying term is also analytic.

As a major application, we prove the analyticity of resonance tongue boundaries in Hill's equation with a small quasi-periodic forcing. Several consequences for the spectrum of Schrödinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schrödinger operators with a small real analytic and quasi-periodic potential has all spectral gaps open and, therefore, it is a Cantor set. Some other applications are included for linear quasi-periodic systems on $so(3,\mathbb{R})$ and $sp(n,\mathbb{R})$.

We consider a topological dynamical system $(\Sigma , \sigma ) on the real line which is topologically conjugate to a topologically mixing one-sided subshift <formula form="inline" disc="math" id="ffm002"><formtex notation="AMSTeX">$(\Sigma_{A}^{+},\sigma_A )$ of finite type with structure matrix $A$. Moreover, we assume that it satisfies some additional conditions so that the map $\sigma$ can have an expanding extension with nice properties in a neighborhood of $\Sigma$. Let $u$ be an eventually positive Lipschitz continuous function on $\Sigma$ and $\zeta_u (s)$ the corresponding dynamical zeta function. We show that $\zeta_u (s)$ has a meromorphic extension in the half-plane $\text{Re }s > -\beta_u$ for some $\beta_u >0$ and its special value at the origin is given by $\zeta_u (0) =1/\text{det}(I-A)$. As an application, we can see that the zeta function $\zeta_Q (s)$ for the two-dimensional dispersing billiard table $Q$ without eclipse has a meromorphic extension in the half-plane $\text{Re }s > -\beta$ for some $\beta>0$ and $\zeta_Q (0)=-1/(J-2)2^{J-1}$ holds, where $J$ is the number of scatterers.

In this paper we discuss whether germs of holomorphic self maps of $\mathbb{C}^n$ can have a dense orbit. We give both a positive and a negative answer depending on how the problem is interpreted more precisely.

The tangent family $f_\lambda(z)=\lambda\tan z$ ($\lambda\in\mathbb{C}\setminus\{0\}, z\in\mathbb{C}$) is considered. It follows from Kotus and Urbański (Math. Ann.324 (2002), 619–656) that the function ascribing to each parameter $\lambda$ the Hausdorff dimension of the Julia set of $f_\lambda$ is continuous at all hyperbolic parameters $\lambda$. Now, we prove that the hyperbolic dimension of the Julia set at each parameter $\lambda_0$ that is a virtual center of a hyperbolic component (in the sense of Keen and Kotus (Conf. Geom. Dyn.1 (1997), 28–57)) is equal to the limit of hyperbolic dimensions (which are also equal to ordinary Hausdorff dimensions) of the Julia sets at hyperbolic parameters $\lambda$ canonically approaching $\lambda_0$ within this component. It is also shown that the Hausdorff dimension of the Julia set of $f_{\lambda_0}$ is strictly larger than this limit.

This paper addresses the following topics relating to the structure of hyperbolic sets: first, hyperbolic sets that are not contained in locally maximal hyperbolic sets; second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a four-dimensional manifold. The third example is volume-preserving and constructed on $\mathbb{R}^4$. We show that every hyperbolic set is included in a hyperbolic set with a Markov partition. In addition, we describe a condition that ensures a hyperbolic set is included in a locally maximal hyperbolic set.