Table of contents

We call a Markov partition of a two-dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular, there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n).

We present a framework for the study of q-difference equations satisfied by q-semi-classical orthogonal systems. As an example, we identify the q-difference equation satisfied by a deformed version of the little q-Jacobi polynomials as a gauge transformation of a special case of the associated linear problem for q-P_VI. We obtain a parametrization of the associated linear problem in terms of orthogonal polynomial variables and find the relation between this parametrization and that of Jimbo and Sakai.

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron–Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds and demonstrate L¹ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.

We study the geometry of the action of on the projective line in order to present a new and simpler proof of the Herman–Avila–Bochi formula. This formula gives the average Lyapunov exponent of a class of 1-families of -cocycles.

When a contractive map is forced by a chaotic discontinuous system, the asymptotic response function that defines the attracting invariant set can be highly irregular. In this context, it is natural to ask whether the invariant distributions of the base and factor systems share the same characteristics and in particular, whether the factor distribution of an absolutely continuous measure in the base can be absolutely continuous. Here, we address this question in a basic example of linear real contractions forced by (generalized) baker's maps and we prove absolute continuity for almost every value of the factor contraction rate.

We study graph-directed iterated function systems of finite type. We show that such an IFS of finite type induces another graph-directed IFS of finite type where every strongly connected component satisfies the open set condition. We introduce the notions of topological and geometric weak separation properties, and summarize the relationship between the different separation conditions. For the induced IFS, the similarity, growth, box and Hausdorff dimensions coincide. Finally we prove that the generalized finite-type condition for graphs implies the geometric weak separation property.

This paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of quadratic codimension-four centres Q₄. Gavrilov and Iliev set an upper bound of eight for the number of limit cycles produced from the period annuli around the centre. Based on Gavrilov–Iliev's proof, we prove in this paper that the perturbed system has at most five limit cycles which emerge from the period annuli around the centre. We also show that there exists a perturbed system with three limit cycles produced by the period annuli of Q₄.

A general model for the description of, e.g., an extensible beam is studied, incorporating weak, viscous and strong as well as Balakrishnan–Taylor damping. Convergence of a sequence of approximate solutions, resulting from a time discretization scheme, towards a weak solution is shown. This also proves the existence of a weak solution.

We present some numerical findings concerning the nature of the blowup versus global existence dichotomy for the focusing cubic nonlinear Klein–Gordon equation in three dimensions for radial data. The context of this study is provided by the classic paper by Payne and Sattinger (1975 Israel J. Math.22 273–303), as well as the recent work by Nakanishi and Schlag (2010 J. Diff. Eqns arXiv:1005.4894). Specifically, we numerically investigate the boundary of the forward scattering region. While the results of (2010 J. Diff. Eqns arXiv:1005.4894) guarantee that this boundary is smooth at energies which are near the ground state energy, it is currently unknown whether or not it continues to be a smooth manifold at higher energies. While we do not find convincing evidence of either smoothness or singularity formation, our numerical work does indicate that at larger energies the boundary becomes much more complicated than at energies near that of the ground state.

In this article, we show that on certain Gatzouras–Lalley carpets, more than one ergodic measure exists with full Hausdorff dimensions. This gives a negative answer to a conjecture of Gatzouras and Peres (1997 Ergod. Theory Dyn. Syst.17 147–67.)

A de Sole, V G Kac and M Wakimoto have recently introduced a new family of compatible Hamiltonian operators of the form H^(N,0) = D² ∘ ((1/u) ∘ D)²ⁿ ∘ D, where N = 2n + 3, n = 0, 1, 2, ..., u is the dependent variable and D is the total derivative with respect to the independent variable. We present a differential substitution that reduces any linear combination of these operators to an operator with constant coefficients and linearizes any evolution equation which is bi-Hamiltonian with respect to a pair of any nontrivial linear combinations of the operators H^(N,0). We also give the Darboux coordinates for H^(N,0) for any odd N ⩾ 3.

Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined discontinuous systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with one boundary in state space and become virtual, and, in the one-dimensional case, the map becomes continuous. Depending on the properties of the map near the codimension-two bifurcation point, there exist different scenarios regarding how the infinite number of periodic orbits are born, mainly the so-called period adding and period incrementing. In our work we prove that, in order to undergo a big bang bifurcation of the period incrementing type, it is sufficient for a piecewise-defined one-dimensional map that the colliding fixed points are attractive and with associated eigenvalues of different signs.

In part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Hölder space of positive exponent for any positive time. In part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity of the flow map.

We study a singular version of the incompressible two-dimensional Navier–Stokes (NS) system on a flat cylinder , with Neumann conditions for the vorticity and a vorticity production term on the boundary to restore the no-slip boundary condition for the velocity . The problem is formulated as an infinite system of coupled ordinary differential equations (ODEs) for the Neumann Fourier modes. For a general class of initial data we prove existence and uniqueness of the solution, and equivalence to the usual NS system. The main tool in the proofs is a suitable decay of the modes, obtained by the explicit form of the ODEs. We finally show that the resulting expansions of the velocity u and of its first and second space derivatives converge and define continuous functions up to the boundary.