TY - JOUR

T1 - Almost periodic structures and the semiconjugacy problem

AU - Aliste-Prieto, J.

AU - Jäger, T.

N1 - Funding Information:
J.A.-P. acknowledges support from Conicyt Chile (Fondecyt Project 3100097) and T.J. acknowledges support from the German Research Council (Emmy-Noether-Grant Ja 1721/2-1).

PY - 2012/5/1

Y1 - 2012/5/1

N2 - The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost periodic base flows, mathematical quasicrystals or maps of the real line with almost periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a rigid translation. We solve this question in a general setting that includes all the above-mentioned examples and also allows the treatment of scalar differential equations that are almost periodic both in space and time. To that end, we study a certain class of flows that preserve a one-dimensional foliation and show that a semiconjugacy to a minimal translation flow exists if and only if a boundedness condition, concerning the distance of orbits of the flow to those of the translation, holds.

AB - The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost periodic base flows, mathematical quasicrystals or maps of the real line with almost periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a rigid translation. We solve this question in a general setting that includes all the above-mentioned examples and also allows the treatment of scalar differential equations that are almost periodic both in space and time. To that end, we study a certain class of flows that preserve a one-dimensional foliation and show that a semiconjugacy to a minimal translation flow exists if and only if a boundedness condition, concerning the distance of orbits of the flow to those of the translation, holds.

UR - http://www.scopus.com/inward/record.url?scp=84857138209&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2012.01.030

DO - 10.1016/j.jde.2012.01.030

M3 - Article

AN - SCOPUS:84857138209

VL - 252

SP - 4988

EP - 5001

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 9

ER -