The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

Let α∈(0,1) be an irrational, and [0;a1,a2,…] the continued fraction expansion of α. Let Hα,V be the one-dimensional Schrödinger operator with Sturmian potential of frequency α. Suppose the potential strength V >20 and the sequence (ai)i≥1 is bounded. We proceed by developing some new ideas on dimensional theory of Cookie-cutter sets. We prove that the spectral generating bands satisfy the principles of bounded variation and bounded covariation, and then we show that there exists a Gibbs-like measure on the spectrum σ(Hα,V). As an application, we prove that where s* and s* are the lower and upper pre-dimensions. Moreover, if (an)n≥1 is ultimately periodic, then s* =s*.

Mean dimension is an invariant which makes it possible to distinguish between topological dynamical systems with infinite entropy. Extending in part the work of Lindenstrauss we show that if (X,ℤk) has a free zero-dimensional factor then it can be embedded in the ℤk-shift on ([0,1]d)ℤk, where d=[C(k) mdim(X,ℤk)]+1 for some universal constant C(k), and a topological version of the Rokhlin lemma holds. Furthermore, under the same assumptions, if mdim(X,ℤk)=0, then (X,ℤk) has the small boundary property. One of the applications of this theory is related to Downarowicz’s entropy structure, a master invariant for entropy theory, which captures the emergence of entropy on different scales. Indeed, we generalize this invariant and prove the Boyle–Downarowicz symbolic extension entropy theorem in the setting of ℤk-actions. This theorem describes what entropies are achievable in symbolic extensions.

We introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of . Some of our main results are as follows: (i) T is weakly mixing if and only if is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if has zero topological entropy, and T has positive entropy if and only if has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

Given a Polish group G of isometries of a locally compact separable metric space, we prove that each measure-preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove our result, we give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert’s fifth problem plays an important role in establishing this characterization.

Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,ℂ)-reducible is in fact G-reducible modulo two if G=GL(n,ℝ),SL(n,ℝ),Sp(n,ℝ) or O(n) and modulo one if G=U(n) .

Let p and q be probability vectors with the same entropy h. Denote by B(p) the Bernoulli shift indexed by ℤ with marginal distribution p. Suppose that φ is a measure-preserving homomorphism from B(p) to B(q). We prove that if the coding length of φ has a finite 1/2 moment, then σ2p=σ2q, where σ2p=∑ ipi(−log  pi−h)2 is the informational variance of p. In this result, the 1/2 moment cannot be replaced by a lower moment. On the other hand, for any θ<1, we exhibit probability vectors p and q that are not permutations of each other, such that there exists a finitary isomorphism Φ from B(p) to B(q) where the coding lengths of Φ and of its inverse have a finite θ moment. We also present an extension to ergodic Markov chains.

Let X be a non-compact surface (or 2-orbifold) of finite type with a metric of strictly negative curvature. Using an ideal tiling of the universal cover of X, we give a symbolic description of the recurrent geodesics on X. This extends Series’s method of coding geodesics on the modular surface.

We extend a theorem of Lotz, which says that any Markov operator T acting on C(X) such that T* is mean ergodic and all invariant measures have non-meager supports must be quasi-compact, to Lotz–Räbiger nets.

We consider a piecewise smooth expanding map on an interval which has two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs. The result is generalized to the case of finitely many invariant components.

We construct a product measure under which the shift on {0,1}ℤ is a type III1 transformation.

We show that any direction in the plane occurs as the unique non-expansive direction of a ℤ2 action, answering a question of Boyle and Lind. In the case of rational directions, the subaction obtained is non-trivial. We also establish that a cellular automaton acting on a subshift can have zero Lyapunov exponents and at the same time act sensitively; and, more generally, for any positive real θ there is a cellular automaton acting on an appropriate subshift with λ+=−λ−=θ.

In this article we prove the existence and uniqueness of equilibrium states for the potential and the class of non-uniformly hyperbolic horseshoes which was introduced in Rios [Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity14 (2001), 431–462]. We show that the pressure t↦𝒫(t) for −tlog Ju is real-analytic on . We give the exact equations of the two asymptotes to the graph of 𝒫(t) at ±∞ and we prove that these non-uniformly hyperbolic horseshoes do not have measures which minimize the unstable Lyapunov exponent.

Given an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

In this paper we establish extreme value statistics for observations on a class of hyperbolic systems: planar dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. In particular, we show that for time series arising from Hölder observations on these systems which are maximized at generic points the successive maxima of the time series are distributed according to the corresponding extreme value distributions for independent identically distributed processes. These results imply an exponential law for the hitting and return time statistics of these dynamical systems.

This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that established by Bergelson, Host and Kra for one single transformation. We use the machinery of ‘magic systems’ established recently by B. Host for the proof.

This paper presents an example of Riemann surface lamination with at least two ergodic invariant measures. The generic leaves for those measures are of different growth and have different numbers of ends.

We extend the renormalization operator introduced in [A. de Carvalho, M. Martens and M. Lyubich. Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121(5/6) (2005), 611–669] from period-doubling Hénon-like maps to Hénon-like maps with arbitrary stationary combinatorics. We show that the renormalization picture also holds in this case if the maps are taken to be strongly dissipative. We study infinitely renormalizable maps F and show that they have an invariant Cantor set 𝒪 on which F acts like a p-adic adding machine for some p>1. We then show, as for the period-doubling case in the work of de Carvalho, Martens and Lyubich [Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121(5/6) (2005), 611–669], that the sequence of renormalizations has a universal form, but that the invariant Cantor set 𝒪 is non-rigid. We also show that 𝒪 cannot possess a continuous invariant line field.

We study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝd on a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝd. In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role both in additive combinatorics and in ergodic theory. Here we introduce a higher-order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.