Title: The spectrum of coupled map lattices: beyond the first gap
Speaker: Professor Viviane Baladi
Speaker Info: University of Geneva
Coupled map lattices are an interesting but tractable model of infinite-dimensional dynamics. No previous knowledge about these objects is required to follow this seminar, which will not be as technical as the following abstract could indicate:Date: Tuesday, April 14, 1998
Following Bricmont-Kupiainen, we consider weakly coupled analytic expanding circle maps on the d-dimensional (d >= 1) lattice, with small coupling strength $\epsilon$ and coupling between two sites decaying exponentially with the distance. We are concerned with the spectrum of the associated (Perron-Frobenius) transfer operators. In a previous work (joint with Degli Esposti, Isola, Jarvenpaa, Kupiainen, to appear J. Math. Pures. Appl. 1998) we had constructed Banach spaces of densities with respect to the coupled SRB measure, and we had applied perturbation theory to the difference of the normalised coupled and uncoupled transfer operators, obtaining localisation of the full spectrum of the coupled operator (i.e., the first spectral gap - which gives exponential decay of time-correlations for a larger class of observables than those previously considered - and beyond). As a side-effect, we had shown that the whole spectra of the truncated (finite-dimensional) coupled transfer operators are localised in a neighbourhood of the truncated uncoupled spectra, uniformly in the spatial size.
A new kernel representation has been introduced in 1997 by Fischer and Rugh, who were able to use a more efficient polymer expansion to get a spectral gap in any lattice dimension and for a larger Banach space of observables. We (joint work in progress with Rugh) are presently trying to combine all the above techniques to study the joint spectrum of the time-transfer operator and the spatial translations.