The Involution Kernel and the Dual Potential for Functions in the Walters’ Family

• Autores: L.Y. Hataishi, A. O. Lopes
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 1, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • First, we set a suitable notation. Points in {0, 1}Z−{0} = {0, 1}N × {0, 1}N = − × +, are denoted by (y|x) = (..., y2, y1|x1, x2, ...), where (x1, x2, ...) ∈ {0, 1}N, and (y1, y2, ...) ∈ {0, 1}N. The bijective map σ (..., ˆ y2, y1|x1, x2, ...) = (..., y2, y1, x1|x2, ...) is called the bilateral shift and acts on {0, 1}Z−{0} . Given A : {0, 1}N = + → R we express A in the variable x, like A(x). In a similar way, given B : {0, 1}N = − → R we express B in the variable y, like B(y). Finally, given W : − × + → R, we express W in the variable (y|x), like W(y|x). By abuse of notation, we write A(y|x) = A(x) and B(y|x) = B(y). The probability μA denotes the equilibrium probability for A : {0, 1}N → R. Given a continuous potential A : + → R, we say that the continuous potential A∗ : − → R is the dual potential of A, if there exists a continuous W : − × + → R, such that, for all (y|x) ∈ {0, 1}Z−{0} A∗(y) = A ◦ ˆσ −1 + W ◦ ˆσ −1 − W (y|x).

    We say that W is an involution kernel for A. It is known that the function W allows to define a spectral projection in the linear space of the main eigenfunction of the Ruelle operator for A. Given A, we describe explicit expressions for W and the dual potential A∗, for A in a family of functions introduced by P. Walters. Denote by θ :

    − × + → − × + the function θ (..., y2, y1|x1, x2, ...) = (..., x2, x1|y1, y2, ...).

    We say that A is symmetric if A∗(θ (x|y)) = A(y|x) = A(x). We present conditions for A to be symmetric and to be of twist type. It is known that if A is symmetric then μA has zero entropy production.

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