A Spectral Gap for a One-dimensional Lattice of Coupled Piecewise Expanding Interval Maps

Abstract

We study one-dimensional lattices of weakly coupled piecewise expand- ing interval maps as dynamical systems. Since neither the local maps need to have full branches nor the coupling map needs to be a homeomorphism of the infinite dimensional state space, we cannot use symbolic dynamics or other techniques from statistical mechanics. Instead we prove that the transfer operator of the infinite dimensional system has a spectral gap on suitable Banach spaces generated by mea- sures with marginals that have densities of bounded variation. This implies in par- ticular exponential decay of correlations in time and space.