No abstract

We are concerned with the tiling flow T associated to a substitution φ over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue λ > 1. The motivation is provided by the (still open) conjecture asserting that T has pure discrete spectrum for any such φ. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual R d -1 -action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as sufficient conditions only-but they all emerge from a unified approach based on the new device: the strand space F φ of φ. The proof of the necessity hinges on determination of the discrete spectrum of T as that of the associated Kronecker toral flow.

No abstract

We prove that every convex polygon in ℝ2with vertices in ℚ2is a rotation set for some isotopic to identity homeomorphism of the two dimensional terms.

We investigate periodic and chaotic solutions of Hamiltonian systems in R 4 which arise in the study of stationary solutions of a class of bistable evolution equations. Under very mild hypotheses, variational techniques are used to show that, in the presence of two saddle-focus equilibria, minimizing solutions respect the topology of the configuration plane punctured at these points. By considering curves in appropriate covering spaces of this doubly punctured plane, we prove that minimizers of every homotopy type exist and characterize their topological properties.