We investigate the β-expansion of an algebraic number in an algebraic base β. Using tools from Diophantine approximation, we prove several results that may suggest a strong difference between the asymptotic behaviour of eventually periodic expansions and that of non-eventually periodic expansions.

This paper gives a direct and elementary proof of the fact that under hypotheses of Tonelli type, solutions to the basic problem in the calculus of variations are Lipschitz when the Lagrangian is autonomous. This fact was first proved by Clarke and Vinter in 1985, using other methods.

A locally free action of a Lie group is parameter rigid if for each other action with the same orbit foliation there exists a orbit-preserving diffeomorphism which conjugates the action to a reparametrization of the other by an automorphism of the Lie group. We show that for actions of the Heisenberg groups, if the first leafwise cohomology group of the orbit foliation is isomorphic to the first cohomology of the Lie algebra of the group, then the action is parameter rigid. Using this, we give examples of parameter rigid actions for all of the Heisenberg groups.

Given a semigroup of surjective local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. We also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

We develop further our study of quasiconformal Anosov flows in our previous (Y. Fang. Smooth rigidity of uniformly quasiconformal Anosov flows. Ergod. Th. & Dynam. Sys.24 (2004), 1–23). For example, we prove the following result: Let φ be a transversely symplectic Anosov flow with dim  Ess≥2 and dim  Esu≥2. If φ is quasiconformal, then it is, up to finite covers, orbit equivalent either to the suspension of a symplectic hyperbolic automorphism of a torus or to the geodesic flow of a closed hyperbolic manifold.

We augment the conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

A dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of , or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

We present the quadratic Weyl sums with θ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[0,1) for which the sequence , n=1,2,… , is dense in .

Let T be an orientation-preserving Lipschitz expanding map of the circle . A pre-image selector is a map with finitely many discontinuities, each of which is a jump discontinuity, and such that τ(x)∈T−1(x) for all . The closure of the image of a pre-image selector is called a flower and a flower with p connected components is called a p-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given p-flower is shown to be of codimension p in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function f has a maximizing measure S which is Sturmian (i.e. is carried by a 1-flower), it is shown that f can be Lipschitz flattened on some 1-flower carrying S.

We completely classify the Kubo–Martin–Schwinger (KMS) states for the gauge action on a C*-algebra associated with a rational function R introduced in our previous work. The gauge action has a phase transition at β=log deg R. We can recover the degree of R, the number of branched points, the number of exceptional points and the orbits of exceptional points from the structure of the KMS states. We also classify the KMS states for C*-algebras associated with some self-similar sets, including the full tent map and the Sierpinski gasket by a similar method.

We give a surface for which the Ricci flow applied to the metric will increase the topological entropy of the geodesic flow. Specifically, we first adapt the Melnikov method to apply to a Ricci flow perturbation and then we construct a surface which is closely related to a surface of revolution, but does not quite have rotational symmetry. This is done by adapting the Liouville metric representation of a surface of revolution. The final steps of the Melnikov method require numerical integration.

A syzygy in the three-body problem is a collinear instant. We prove that, with the exception of Lagrange’s solution, every solution to the zero angular momentum, Newtonian three-body problem suffers syzygies. The proof works for all mass ratios.

We show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2×2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge–Kutta–Fehlberg algorithm. From this, we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.

Let be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the -action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group . This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on .

We relate Kellendonk and Putnam’s pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with values in any Abelian group. It also provides an easy proof of Kellendonk and Putnam’s original theorem relating PE cohomology to the Čech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.