Spatially Extended Circle Maps: Monotone Periodic Dynamics of Functions with Linear Growth
We introduce and study monotone periodic mappings acting on real functions with linear growth. These mappings represent the nonlinear dynamics of extended systems governed by a diffusive interaction and a periodic potential. They can be viewed as infinite-dimensional analogues of lifts of circle maps. Our results concern the existence and uniqueness of a rotation number and the existence of travelling waves. Moreover, we prove that the rotation number depends continuously on the mapping and we obtain a symmetry condition for this number to vanish. The results are applied to two classes of examples in population dynamics and in condensed matter physics.
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