We show that the ordered Matsumoto K0-group of a substitutive system is derivable from the ordered Čech cohomology of a simple ‘augmentation’ of the tiling space associated with the substitution. Besides supplying a topological interpretation of the Matsumoto K0-group, this leads to finer invariants of the flow equivalence class of a substitutive system.

We study aperiodic substitution dynamical systems arising from non-primitive substitutions. We prove that the Vershik homeomorphism φ of a stationary ordered Bratteli diagram is topologically conjugate to an aperiodic substitution system if and only if no restriction of φ to a minimal component is conjugate to an odometer. We also show that every aperiodic substitution system generated by a substitution with nesting property is conjugate to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitution system is recognizable. The classes of m-primitive substitutions and derivative substitutions associated with them are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

Assume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of , is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

A measure-preserving isomorphism α of a probability space (Ω,ℱ,μ) generates an algebraic automorphism of the algebra of bounded linear operators in 𝕃p(Ω,ℱ,μ). An ergodic theorem for is proved.

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of 𝕋4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

Let 0<θ<1 be an irrational number of bounded type. We prove that for any map in the family (e2πiθz+αz2)ez, α∈ℂ, the boundary of the Siegel disk, with fixed point at the origin, is a quasi-circle passing through one or both of the critical points.

We prove that general topological dynamical systems on metric Cantor spaces can be embedded into cellular automata defined on Cayley graphs. Furthermore, we prove that it is possible to construct this embedding in such a way that topological entropy is preserved.

We shall prove that C1-robustly expansive codimension-one homoclinic classes are hyperbolic.

Let Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

Purely numerical methods do not always provide an accurate way to find all the global solutions to nonlinear ordinary differential equations on infinite intervals. For example, finite-difference methods fail to capture the asymptotic behavior of solutions, which might be critical for ensuring global existence. We first show, by way of a detailed example, how asymptotic information alone provides significant insight into the structure of global solutions to a nonlinear ordinary differential equation. Then we propose a method for providing this missing asymptotic data to a numerical solver, and show how the combined approach provides more detailed results than either method alone.

In this article we construct a new framework for the study of polygonal billiards. To every polygonal billiard we associate a holomorphic foliation on C2. The dynamics of the billiard ball is linked to the directional flow of the complex vector field defining the associated holomorphic foliation.

We formulate an abstract fluctuation theorem which sheds light on mathematical relations between the fluctuation theorems of Bochkov and Kuzovlev [Contribution to the general theory of thermal fluctuations in nonlinear systems. Sov. Phys.–JETP45 (1977), 125] and Jarzynski [Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys.98 (2001), 77–102] on the one hand, and those of Evans and Searles [Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50 (1994), 1645–1648] and Gallavotti and Cohen [Dynamical ensembles in stationary states. J. Stat. Phys.80 (1995), 931–970] on the other.

We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a ‘general Sierpiński carpet’ was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpiński carpet represented by a shift of finite type. We give some conditions under which a general Sierpiński carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.