We consider horizontal iterated function systems in the Heisenberg group $\mathbb{H}^1$, i.e. collections of Lipschitz contractions of $\mathbb{H}^1$ with respect to the Heisenberg metric. The invariant sets for such systems are so-called horizontal fractals. We study questions related to connectivity of horizontal fractals and regularity of functions whose graph lies within a horizontal fractal. Our construction yields examples of horizontal BV (bounded variation) surfaces in $\mathbb{H}^1$ that are in contrast with the non-existence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim (Rectifiable sets in metric and Banach spaces. Math. Ann.318(3) (2000), 527–555).

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approach, pertaining to the fundamentals of ergodic theory, which in particular does not require Pesin theory. This allows us to show, for those systems, that the Hausdorff dimension of a (non-ergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition. We also establish corresponding results for non-uniformly hyperbolic diffeomorphisms on surfaces, now using Pesin theory. To the best of our knowledge, the formula for the Hausdorff dimension of an invariant measure along an ergodic decomposition is new even in the case of uniformly hyperbolic dynamics.

We show that given an adding machine of type $\alpha$, for a dense set of parameters $s$ in the interval $[\sqrt2,2]$, if $f$ is the tent map with slope $s$, then the restriction of $f$ to the closure of the orbit of the turning point is topologically conjugate to the adding machine map of type $\alpha$.

In this paper we obtain different types of random ergodic theorems for dynamical systems or continuous semi-flows. These results recover and extend previous works on dynamical systems and are completely new in the case of semi-flows. The proofs are based on uniform estimates on random almost periodic polynomials that we obtained very recently and on an improvement of a tool introduced by Talagrand in 1995 and further developed by Fernique in 1996. In the course of this, we partially recover results of Marcus and Pisier from the 1980s on almost sure uniform convergence of random almost periodic series.

Let $(Y_k)$ be an integrable sequence of iid random variables defined on the probability space $(Y,\mathcal F, \mu)$. We prove that there exists a subset $Y^{*}\subset Y$ of full measure such that for each $y\in Y^{*}$ the following holds: for every integrable iid sequence $(X_k)$ on a probability space $(X,\Sigma,m)$, the series

$$\lim_{n\to\infty}\sideset{}{'}\sum_{k=-n}^{n}\frac{Y_k(y)X_k(x)}{k}$$

converges for almost every $x\in X$.

It is proved that all special flows over a rotation by an irrational $\alpha$ with bounded partial quotients and under f which is piecewise absolutely continuous with a non-zero sum of jumps are mildly mixing. Such flows are also shown to enjoy a condition that emulates the Ratner condition introduced in M. Ratner (Horocycle flows, joinings and rigidity of products. Ann. of Math.118 (1983), 277–313). As a consequence we construct a smooth vector-field on $\mathbb{T}^2$ with one singularity point for which the corresponding flow $(\varphi_t)_{t\in\mathbb{R}}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on which $(\varphi_t)_{t\in\mathbb{R}}$ is mildly mixing and is spectrally disjoint from all mixing flows.

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 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.

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.

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.

Let $h:D\rightarrow {\mathbb R}^2$ be a homeomorphism of a disk into the plane. Let $C\subset {\rm int}\,D$ be a Cantor set such that $(C,h|_C)$ is an adding machine. We give a condition under which there are points outside of C which have their entire orbits contained in D.

We introduce the notions of a rational point, a rational sub-nilmanifold and a rational homomorphism of compact nilmanifolds and describe elementary properties thereof. We also show that the closure of the orbit of a point of a nilmanifold $X$ under a polynomial action of the group $\mathbb{Z}^{d}$ by translations is, up to a shift, a union of rational sub-nilmanifolds of $X$.

It is a safe conjecture that most (not necessarily periodic) two-dimensional Lorentz gases with finite horizon are recurrent. Here we formalize this conjecture by means of a stochastic ensemble of Lorentz gases, in which independent identically distributed random scatterers are placed in each cell of a co-compact lattice in the plane. We prove that the typical Lorentz gas, in the sense of Baire, is recurrent, and give results in the direction of showing that recurrence is an almost sure property (including a zero--one law that holds in every dimension). A few toy models illustrate the extent of these results.

We consider a class of expanding maps of the 2-torus of the form $f(x,y)=(a(x,y), b(y))$ that are C$^2$-perturbations of linear ones. In that class we consider invariant sets $\Lambda$ possessing a simple Markov structure, and show there exist ergodic invariant measures supported on $\Lambda$ with Hausdorff dimension arbitrarily close to the Hausdorff dimension of $\Lambda$. For f and $\Lambda$ as above we also show that the Birkhoff exceptional set has full Hausdorff dimension.

Measures associated to (compact, non-singular) hyperbolic surface laminations are studied. The main results of this work relate harmonic measures and measures invariant under the laminated geodesic and horocycle flows.

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275). More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally) Lipschitz functions from $B$ to $\mathbb{R}^{m}$ that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function $f \in M$, the Hausdorff dimension of $f(X)$ is at least $\min\{ m, d / (1 + \tau) \}$. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on $X$. The factor $1 / (1 + \tau)$ can be improved to $1 / (1 + \tau / 2)$ if $B$ is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when $\tau = 0$. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case $\tau > 0$.

This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563–603). We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere $\overline{\mathbb C}$ of degree two or greater, the Julia set of any subsemigroup of G is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic, then the fiberwise Julia sets are porous, and the dynamics is weakly rigid. Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise maps are polynomials, then under some conditions, the fiberwise basins of infinity are John domains. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on $\overline{\mathbb C}$) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

Generalizing a three-dimensional result of Doering, we prove that a robustly transitive vector field defined on a compact manifold of any dimension greater than three admits no fixed point.