We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type.

A rational map f is called geometrically finite if every critical point contained in its Julia set is eventually periodic. If a perturbation of f into another geometrically finite rational map is horocyclic and preserves the critical orbit relations with respect to the Julia set of f, then we can construct a semiconjugacy or a topological conjugacy between their dynamics on the Julia sets.

This self-contained paper is part of a series (Groups of homeomorphisms of one-manifolds I: actions of nonlinear groups. Preprint, 2001; Group actions on one-manifolds II: extensions of Hölder's Theorem. To appear in Trans. Amer. Math. Soc.) seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante and Thurston proved that every nilpotent subgroup of Diff2(S1) is abelian. One of our main results is a sharp converse: Diff1(S1) contains every finitely generated, torsion-free nilpotent group.

An action of $\mathbb{Z}^k$ is associated to a higher rank graph $\Lambda$ satisfying a mild assumption. This generalizes the construction of a topological Markov shift arising from a non-negative integer matrix. We show that the stable Ruelle algebra of $\Lambda$ is strongly Morita equivalent to $C^*(\Lambda)$. Hence, if $\Lambda$ satisfies the aperiodicity condition, the stable Ruelle algebra is simple, stable and purely infinite.

We find an explicit example of a Markov odometer action which is not orbit-equivalent to any product odometer action. We make use of some techniques from Krieger's construction of an action of non-product type arising from a flow. However, his example is not realized as an odometer action with an explicit measure, nor does it arise from a Markov measure.

For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the $L^q$-spectrum form a Legendre transform pair. The main difficulty comes from the possible existence of first-order phase transition points, that is, points where the $L^q$-spectrum is not differentiable. We give examples of weak Gibbs measure with phase transition, including the so-called Erdös measure.

Nous étendons des théorèmes sur le spectre de Markoff et de Lagrange connus dans le cas rationnel au cas analogues imaginaires quadratiques. Pour ce faire, nous nous ramenons à un problème de géométrie hyperbolique.

In this paper, random expansions to non-integer bases \beta >1 are studied. For \beta's satisfying \beta^2=n\beta +k (with 1\leq k\leq n) and \beta^n=\beta^{n-1}+\dotsb + \beta +1 the ergodic properties of such expansions are described.

Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi:X\to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects to \nu and has maximal entropy among all measures in the fiber. However, there is an explicit bound on the number of such maximal entropy preimages.

We use matrices over formal power series to represent irreducible and positive recurrent Markov chains and identify a natural class of good finitary isomorphisms (magic word isomorphisms) as those arising from elementary matrix operations. This extends the positive K-theory framework which has been used for other classification problems in symbolic dynamics.

This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X-X is locally finite. Such sets are characterized by their patch-counting function N_X (T) of radius T being finite for all T. We formulate conjectures relating slow growth of the patch-counting function N_X (T) to the set X having a non-trivial translation symmetry.

A Delone set X of finite type is repetitive if there is a function M_X(T) such that every closed ball of radius M_X(T) + T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with \mathbb{R}^n-action. There is a lower bound for M_X(T) in terms of N_X(T), namely M_X(T) \ge c( N_X (T))^{1/n} for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for M_X(T) purely in terms of N_X(T). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M_X (T). For example, the function M_X (T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if M_X (T) = O(T) as T \to \infty and is densely repetitive if M_X (T) = O(N_X(T))^{1/n} as T \to \infty. We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure. In the reverse direction, we construct a repetitive Delone set X in \mathbb{R}^n which has M_X (T) = O(T (\log T)^{2/n} (\log\log\log T )^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of ‘perfectly ordered quasicrystals’.

Recently, the stretched exponential decay of multiple correlations in a periodic Lorentz gas has been used to show the convergence of a series of correlations which has a physical interpretation as the fourth-order Burnett coefficient, a generalization of the diffusion coefficient. Here the result is extended to include all higher-order Burnett coefficients and a plausible argument is given that the expansion constructed from the Burnett coefficients has a finite radius of convergence.

We study the relation between topological entropy and the dispersion of preimages. Symbolic dynamics plays a crucial role in our investigation. For forward expansive maps, we show that the two pointwise preimage entropy invariants defined by Hurley agree with each other and with topological entropy, and are reflected in the growth rate of the number of preimages of a single point, called a preimage growth point for the map. We extend this notion to that of an entropy point for a system, in which the dispersion of preimages of an $\varepsilon$-stable set measures topological entropy. We show that for maps satisfying a weak form of the specification property, every point is an entropy point and that every asymptotically h-expansive homeomorphism (in particular, every smooth diffeomorphism of a compact manifold) has entropy points. Examples are given of maps in which Hurley's invariants differ and of homeomorphisms with no entropy points.

Two measure-preserving dynamical systems are weakly disjoint if some pointwise convergence property is satisfied by ergodic averages on their direct product. Disjointness implies weak disjointness. We start studying this new concept, both by stating some general properties and by giving various examples.

We consider the standard family (or Arnold family) of circle maps given by f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}, for x,\alpha\in [0,2\pi), \beta \in (0,1) and its complexification F_{\alpha,\beta}(z)=z e^{i\alpha} \exp [\frac12\beta(z-\frac{1}{z})]. If f_{\alpha,\beta} is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of F_{\alpha,\beta}. Given an irrational rotation number \theta, the parameters (\alpha,\beta) such that f_{\alpha, \beta} has rotation number \theta form a curve T_\theta in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if \theta is a Brjuno number, the curve T_\theta can be parametrized real-analytically by the modulus of the Herman ring, from \beta=0 up to a point (\alpha_0,\beta_0) with \beta_0 \leq 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables37 (1998), 67–98) we show that for a certain set of angles \theta \in \mathcal{B} \setminus \mathcal{H}, the point \beta_0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of F_{\alpha, \beta}.

For each positive integer n we use the concept of ‘admissible arrays on n symbols’ to define a set of positive integers $Q(n)$ which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In earlier joint work with M. Scheutzow, it was shown that the set $Q(n)$ is intimately connected to the set of periods of periodic points of classes of non-expansive nonlinear maps defined on the positive cone in $\mathbb{R}^n$. In this paper we continue the characterization of $Q(n)$ and present precise asymptotic estimates for the largest element of $Q(n)$. For example, if $\gamma(n)$ denotes the largest element of $Q(n)$, then we show that $\lim_{n \to \infty} (n\log n)^{-1/2}\log \gamma(n) = 1$. We also discuss why understanding further details about the fine structure of $Q(n)$ involves some delicate number theoretical issues.

We consider a class of billiard tables (X,g), where X is a smooth compact manifold of dimension two with smooth boundary \partial X and g is a smooth Riemannian metric on X, the billiard flow of which is completely integrable. The billiard table (X,g) is defined by means of a special double cover with two branched points and it admits a group of isometries G \cong \mathbb{Z}_2 \times\mathbb{Z}_2. Its boundary can be characterized by the string property; namely, the sum of distances from any point of \partial X to the branched points is constant. We provide examples of such billiard tables in the plane (elliptical regions), on the sphere \mathbf{S}^2,on the hyperbolic space \mathbf{H}^2, and on quadrics. The main result is that the spectrum of the corresponding Laplace–Beltrami operator with Robin boundary conditions involving a smooth function K on \partial X uniquely determines the function K, provided that K is invariant under the action of G.

We prove that the general (i.e. possibly aperiodic) Lorenz gas in two dimensions, with finite horizon and non-degenerate geometrical features, is ergodic if it is recurrent. We also give examples of aperiodic recurrent gases.

For a sequence (cn) of complex numbers, the quadratic polynomials $f_{c_n}:=z^2+c_n$ and the sequence (Fn) of iterates $F_n:=f_{c_n}\circ\dotsb \circ f_{c_1}$ are considered. The Fatou set $\mathcal{F}(c_n)$ is defined as the set of all $z\in \hat{\mathbb{C}}: =\mathbb{C}\cup \{\infty\}$ such that (Fn) is normal in some neighbourhood of z, while the complement $\mathcal{J}(c_n)$ of $\mathcal{F}(c_n)$ (in $\hat{\mathbb{C}}$) is called the Julia set. In this paper we discuss the conditions for $\mathcal{J}(c_n)$ to be totally disconnected. A problem posed by Brück is solved.

A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.