We consider weakly coupled analytic expanding circle maps on the lattice $\integer^d$ (for $d\ge 1$), with small coupling strength $\epsilon$ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fr\'echet space on which the operator associated to the full system has a simple eigenvalue at $1$ (corresponding to the SRB measure $\mu_\epsilon$ previously obtained by Bricmont--Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For $d=1$ we also construct Banach spaces of densities with respect to $\mu_\epsilon$ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are $\OO(\epsilon)$-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [BK1].

The spectrum of weakly coupled maps

ISOLA, Stefano;
1998-01-01

Abstract

We consider weakly coupled analytic expanding circle maps on the lattice $\integer^d$ (for $d\ge 1$), with small coupling strength $\epsilon$ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fr\'echet space on which the operator associated to the full system has a simple eigenvalue at $1$ (corresponding to the SRB measure $\mu_\epsilon$ previously obtained by Bricmont--Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For $d=1$ we also construct Banach spaces of densities with respect to $\mu_\epsilon$ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are $\OO(\epsilon)$-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [BK1].
1998
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11581/116848
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact