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By M. Aizenman (Chief Editor)

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193(3), 675–711 (1998) 21. : Coupled map lattice via transfer operators on functions of bounded variation. In: Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks 45, Amsterdam: North-Holland, 1996, pp. 71–80 22. : Mixing for finite systems of coupled tent maps. Tr. Mat. Inst. Steklova 216, pp. 320–326 (1997), Din. Sist. ; translation in Proc. Steklov Inst. Math. 1997, no. 1, 315–332 (1997) 23. : An ergodic theoretic approach to mean field coupled maps.

Stochastic stability of weakly coupled map lattices. Nonlinearity 10, 715–730 (1997) 30. : Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice. Pys. Rev. E, 48, 2528–2535 (1993). 31. : Space-time chaos in chains of weakly interacting hyperbolic mappings. Math. 3, 165-198 (1991) ´ 32. H. Rugh, Coupled maps and analytic function spaces. Ann. Sci. Ecole Norm. Sup. (4) 35(4) 489–535 (2002) 33. : BV -spectral theory for coupled map lattices. Dissertation, Universit¨at Erlangen (2003).

Let ϕ be a function depending only on the variables belonging to a box 0 ∈ I, so that ϕ ◦ T n depends only on the variables in the nw-neighborhood n of 0 . Then µ(ϕ) ˜ = T ∗n (µ˜ − µδ,η )(ϕ) + T ∗n (µδ,η − µ )(ϕ) + µ (ϕ). 2) We must prove that µ(ϕ) ˜ = µ (ϕ). As |T ∗n (µ˜ − µδ,η )| ≤ |µ˜ − µδ,η | ≤ δ with an arbitrary δ > 0, it just remains to show that the second term can be made as small as we like by choosing η and n appropriate. Given a measure µ let µ n be its marginal with respect to the box n .

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