Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

Abstract

In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincaré’s theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.The author acknowledges support from a CONICYT doctoral fellowship and grants: ECOSCONICYT C03EC03, Nucleo Milenio P04-069-F, ANR Crystal Dyn and Basal-CMM

Almost periodic structures and the semiconjugacy problem

“…The following result (which already appears in [AP10]) is again a direct consequence of Theorem 1.1.…”

“…In [JS06], a Poincaré-like classification is given for skew-products on the two-torus over irrational rotations (so-called quasiperiodically forced circle homeomorphisms). For maps on mathematical quasicrystals, a similar classification was obtained in [AP10]. While in both situations strong use is made of the order-preserving properties of the considered systems, corresponding to the existence of a one-dimensional foliation that is preserved by the dynamics, the methods employed are quite different.…”

Irrational rotation factors for conservative torus homeomorphisms

“…Even 130 years after Poincaré's contribution, similar results in this direction are quite rare. In recent years, however, there has been substantial progress on 'topological linearization' in a number of situations and system classes, including skew-products over irrational rotations [3], mathematical quasicrystals [4], reparametrisations of irrational flows [5], area-preserving torus homeomorphisms [6] and dynamics on circle-like continua [7]. Most importantly, some general pattern and methods start to emerge.…”

Topological friction in aperiodic minimal Rm-actions

“…The Birkhoff averages X φ dµ are called rotation vectors and constitute an important topological invariant of the dynamics of f (cf. [6,1]).…”

“…The author thanks the Centre International de Rencontres Mathématiques, Luminy, for its hospitality and expresses his gratitude to José Aliste-Prieto and Jean-Marc Gambaudo whose question about continuity of φ in the context of a minimal tiling flow prompted this note (cf. [1]).…”