The spectrum of weakly coupled map lattices

Abstract

We consider weakly coupled analytic expanding circle maps on the lattice Z d (for d ≥ 1), with small coupling strength and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fréchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure µ 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 µ 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 O( )-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].Typeset by A M S-T E X 1 5

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

“…Exceptions to this failure are repellers of weakly coupled chains of maps with Cantor repelling set [12,13] and specially designed CML for which the coupling operator preserves the uncoupled Markov partition [2,8,14,17,21,35,36,39]. Independently of grammatical issues, proofs of uniqueness of the physical measure in the weak coupling regime (analogue to the uniqueness of the high temperature phase) have been provided using perturbative approaches from the uncoupled limit [1,5,6,16,19,27,28].…”

Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling

“…5) Here ∂ i denotes the partial derivative with respect to x i . 2 It is easy to prove that the set BV Ω := {µ ∈ M(Ω) : Var µ < ∞} consists of measures whose finite dimensional marginals are absolutely continuous with respect to Lebesgue and the density is a function of bounded variation [18]. In addition, such measures have finite entropy density with respect to Lebesgue [19,Corollary 4.1].…”

“…if, for each 1 We use the product topology on Ω. 2 Here and in the sequel all test functions ϕ : Ω → R depend on only finitely many coordinates and are C 1 with respect to these coordinates. 3 See [19] for a careful discussion of bounded variation in the present context and the relevant associated properties.…”

“…So we are not able to construct examples of phase transitions in a class of systems as studied e.g. in [2,3,6,7,8,9,11,25]. …”

Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension

“…Starting with [7] numerous authors contributed to the exploration of ergodic and statistical properties of invariant measures for such systems, see e.g. [1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,17,18,19,20,29,31,32,34,35]. In all these publications the single site maps are hyperbolic or expanding (local) diffeomorphisms of a smooth manifold, and the coupling is modeled by a "diffeomorphism" of the infinite-dimensional state space.…”