Let V be a finite-dimensional vector space over a field ${\mathbb{K}}$ and let G be a countable amenable group. Let $\tau \colon V^G \to V^G$ be a linear cellular automaton. We prove that $\tau$ is surjective if and only if it is pre-injective.

In this article we study the asymptotic dynamics of highly oscillatory solutions for the unbalanced Allen–Cahn equation with a slowly varying coefficient. We describe the underlying structure of these solutions through a function we call the adiabatic profile, which accounts for the asymptotic area covered by the solutions in the phase space. In finite intervals, we construct solutions given any adiabatic profile. In the case of a periodic coefficient we show that the system has chaotic behavior by constructing high-frequency complex solutions which can be characterized by a bi-infinite sequence of real numbers in $[c_1,c_2]\cup\{ 0\}\ (0<c_1<c_2)$.

Let $\Omega$ be an abelian group. A set $R\subset \Omega$ is a set of recurrence if, for any probability measure-preserving system $(X,{\mathcal B},\mu,\{T_g\}_{g\in \Omega})$ and any $A\in {\mathcal A}$ with $\mu(A)>0,\ \mu(A\cap T_g A)>0$ for some $g\in R$. If $(x_i)_{i=1}^\infty$ is a sequence in $\Omega$, the set of its finite sums $\{ x_{i_1}+x_{i_2}+\dotsb + x_{i_k}: i_1<i_2<\dotsb <i_k\}$ is called an IP-set. In Bergelson et al (Erg. Theor. Dyn. Sys.16 (1996), 963–974) it is shown that if $p:{\mathbb Z}^d{\rightarrow} {\mathbb Z}^k$ is a polynomial vanishing at zero and F is an IP-set in ${\mathbb Z}^d$ then $\{ p(n):n\in F\}$ is a set of recurrence in ${\mathbb Z}^k$. Here we extend this result to an analogous family of generalized polynomials, that is functions formed from regular polynomials by iterated use of the greatest integer function, as a consequence of a theorem establishing a much wider class of recurrence sets occurring in any (possibly non-finitely generated) abelian group. While these sets do in a sense have a distinctively ‘polynomial’ nature, this far-ranging class includes, even in ${\mathbb Z}$, such examples as $\big\{\!\sum_{i, j\in \alpha, i<j} 2^i3^j : \alpha\subset \mathbb{N}, 0<|\alpha|<\infty\big\}$, where the connection to conventional polynomials is somewhat distant.

We study transversely Lorentzian foliations of closed 3-manifolds. We classify them under a completeness hypothesis and deduce the dual classification of codimension-one geodesically complete timelike totally geodesic foliations. Also we provide an example of a Lorentzian foliation of a compact 3-manifold which is equicontinuous on a proper dense open subset of the manifold, and hence not transversely complete.

In this paper we study analytic properties of compensator and Dulac expansions in a single variable. A Dulac expansion is formed by monomial terms that may contain a specific logarithmic factor. In compensator expansions this logarithmic factor is deformed. We first consider Dulac expansions when the power of the logarithm is either 0 or 1. Here we construct an explicit exponential scaling in the space of coefficients which in an exponentially narrow horn, up to rescaling and division, leads to a polynomial expansion. A similar result holds for the compensator case. This result is applied to the bifurcation theory of limit cycles in planar vector fields. The setting consists of families that unfold a given Hamiltonian in a dissipative way. This leading part is of Morse type, which leads to the following three cases. The first concerns a Hamiltonian function that is regular on an annulus, and the second a Hamiltonian function with a non-degenerate minimum defined on a disc. In the third case the Hamiltonian function has a non-degenerate saddle point with a saddle connection. The first case, by an appropriate scaling, recovers the generic theory of the saddle node of limit cycles and its cuspoid degeneracies, while the second case similarly recovers the generic theory of limit cycles subordinate to the codimension $k$ Hopf bifurcations $k = 1,2,\dotsc$. The third case enables a novel study of generic bifurcations of limit cycles subordinate to homoclinic bifurcations. We then describe how the above analytic result is applied to bifurcations of limit cycles. For appropriate one-dimensional Poincaré maps, the fixed points correspond to the limit cycles. The fixed point sets (or zero-sets of the associated displacement functions) are studied by contact equivalence singularity theory. The cases where the Hamiltonian is defined on an annulus or a disc can be reduced directly to catastrophe theory. In the third case, the displacement functions are known to have compensator expansions, whose first approximations are Dulac expansions. Application of our analytic result implies that, in an exponentially narrow horn near a homoclinic loop, the bifucation theory of limit cycles again reduces to catastrophe theory.

We prove quantitative equidistribution results for nilflows on compact three-dimensional homogeneous nilmanifolds by a method based on renormalization and invariant distributions for nilflows. As an application we obtain a dynamical proof of quantitative equidistribution results for the sequence P(n) (mod 1), where $P(X)\in \mathbb{R}[X]$ is a quadratic polynomial with generic leading coefficient. Bounds on theta sums, that is, Birkhoff sums of the exponential function along such sequences, were proved by Hardy and Littlewood and in optimal form by Fiedler, Jurkat and Körner.

We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map ${\mathcal I}$ from the parameter space of possible shapes of tiles to H1 of a model tiling space, with values in ${\mathbb R}^d$. Two tiling spaces that have the same image under ${\mathcal I}$ are mutually locally derivable (MLD). When the difference of the images is ‘asymptotically negligible’, then the tiling dynamics are topologically conjugate, but generally not MLD. For substitution tilings, we give a simple test for a cohomology class to be asymptotically negligible, and show that infinitesimal deformations of shape result in topologically conjugate dynamics only when the change in the image of ${\mathcal I}$ is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be topologically weakly mixing.

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 $x\in X$.

We study the dynamics near transverse intersections of stable and unstable manifolds of sheets of symmetric periodic orbits in reversible systems. We prove that the dynamics near such homoclinic and heteroclinic intersections is not $C^1$ structurally stable. This is in marked contrast to the dynamics near transverse intersections in both general and conservative systems, which can be $C^1$ structurally stable. We further show that there are infinitely many sheets of symmetric periodic orbits near the homoclinic or heteroclinic orbits. We establish the robust occurrence of heterodimensional cycles, that is, heteroclinic cycles between hyperbolic periodic orbits of different index, near the transverse intersections. This is shown to imply the existence of hyperbolic horseshoes and infinitely many periodic orbits of different index, all near the transverse intersections.

In this paper, which is a sequel to our previous work (On a general Conley index continuation principle for singular perturbation problems. Ergod. Th. & Dynam. Sys.22 (2002), 729–755) we establish a general singular nested index filtration principle which, in particular, implies a continuation principle for homology index braids and connection matrices in singular perturbation problems.

Theorem 1.1 in [4] is false: the hypothesis $\sum v_n(g)<\infty$ needs to be replaced by a stronger condition. A corrected version of Theorem 1.1 is given below; a counterexample to the original result is given in [5].

In 1993, E. Lesigne proved a polynomial extension of the Wiener–Wintner theorem and asked two questions: does this result have a uniform counterpart and can an assumption of total ergodicity be replaced by ergodicity? The purpose of this article is to answer these questions, the first one positively and the second one negatively.

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.

Let $X$ be a one-sided subshift of finite type on a countable alphabet, and $T:X\to X$ the shift map. If $f:X\to{\mathbb R}$ is continuous, we provide conditions guaranteeing that $f$-maximizing measures exist, and are characterised by the condition that their support lies in a certain compact set.

We prove that for a suitable (open) class of open, smoothly bounded domains in the cotangent bundle of a surface of genus $g \geq 2$ any exact symplectomorphism is homotopic to one which is smooth up to the boundary. In particular, such boundaries are not unseen in the sense of Eliashberg–Hofer. The contact boundaries of these domains have Anosov Reeb flows. The methods employed include the properties of such flows in three dimensions, and analysis of two-dimensional transverse flow maps along the lines of Thurston, and thus appear restricted to the case of three-dimensional contact boundaries.

If S(t) is a C0-semigroup on a Banach space X with a hypercyclic vector x, then x is a hypercyclic vector for each discrete semigroup $\lbrace S(kT)\mid k \in \mathbb{N}\rbrace$ with T > 0 fixed.

We introduce and study, from a combinatorial-topological viewpoint, some semigroups of continuous non-deterministic dynamical systems. Combinatorial stability, i.e. the persistence of the combinatorics of the attractors, is characterized and its genericity established. Some implications on topological (deterministic) dynamics are drawn.

The notion of the maximal pattern complexity of words was introduced by Kamae and Zamboni. In this paper, we obtain an almost exact formula for the maximal pattern complexity $p^*_\alpha(k)$ of Toeplitz words $\alpha$ on an alphabet ${\mathbb A}$ defined by a sequence of coding words $(\eta^{(n)})^\infty\in({\mathbb A}\cup\{?\})^{\mathbb N}(n=1,2,\dotsc)$ including just one ‘?’ in their cycles $\eta^{(n)}$. Using this formula, we characterize pattern Sturmian words (i.e. $p^*_\alpha(k)=2k$ (for all $k$)) in this class. Moreover, we give a characterization of simple Toeplitz words in the sense of Kamae and Zamboni in terms of pattern complexity. In the case where $\eta^{(1)}=\eta^{(2)}=\dotsb$, we obtain the value $\lim_{k\to\infty}p_\alpha^*(k)/k$. We construct a Toeplitz word $\alpha\in{\mathbb A}^{\mathbb N}$ with $\#\A=2$ such that $p^*_\alpha(k)=2^k~(k=1,2,\dotsc)$, while Toeplitz words in our sense always have discrete spectra.

An example of non-unique g-measures is constructed. The function g satisfies a certain summability condition, thereby providing a counterexample to the claim in an earlier paper that all such g have a unique g-measure.

In this paper we show that, for every Choquet simplex $K$ and for every $d>1$, there exists a ${\mathbb Z}^d$-Toeplitz system whose set of invariant probability measures is affine homeomorphic to $K$. Then, we conclude that $K$ may be realized as the set of invariant probability measures of a tiling system $(\Omega_T,{\mathbb R}^d)$.