By M. Aizenman (Chief Editor)
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We recall the notion of a symbolic (semi)-flow [5, 19]. Suppose that σ : X+ → X + is an aperiodic one-sided subshift of finite type. Fix θ ∈ (0, 1). Define the metric dθ (x, y) = θ N , where N is the largest positive integer such that xi = yj for all i < N . Define the Hölder space Fθ (X + ) consisting of continuous functions v : X+ → R that are Lipschitz with respect to this metric, with Lipschitz constant |v|θ . Let µ be an equilibrium measure on X + corresponding to a Hölder potential in Fθ (X + ).
Hence |U ∗ f |∞ ≤ |f |∞ . Finally, compute that p−1 p−1 |U ∗ f |p = |U ∗ f |p−1 |U ∗ f | ≤ |U ∗ f |∞ |U ∗ f |1 ≤ |f |∞ C. 60 I. Melbourne, A. Török Lemma 1. Let (Y, m) be a probability space and T : Y → Y be a measure preserving transformation. Define U ∗ : L2 → L2 as in Proposition 1. Let φ : Y → R be in L∞ with Y φ dm = 0. Fix n > 2, and suppose that there is a constant C (depending on φ and n) such that Yφ (ψ ◦ T j ) dm ≤ C |ψ|∞ , jn (2) for all ψ ∈ L∞ and j ≥ 1. Then φ = φ + χ ◦ T − χ , where φ and χ lie in Lp , for all p < n, and U ∗ φ = 0.
E. 16 (8,9), 1223–1253 (1991) 12. : Vortex condensation in the Chern–Simons–Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995) 13. : A special class of stationary flows for two dimensional Euler equations: A statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992) 14. : A special class of stationary flows for two dimensional Euler equations: A statistical mechanics description, part II. Commun. Math. Phys. 174, 229–260 (1995) 46 D. Bartolucci, G. Tarantello 15.
Communications in Mathematical Physics - Volume 229 by M. Aizenman (Chief Editor)