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

First, we set a suitable notation. Points in { 0 , 1 } Z - { 0 } = { 0 , 1 } N × { 0 , 1 } N = Ω - × Ω + , are denoted by ( y | x ) = ( . . . , y 2 , y 1 | x 1 , x 2 , . . . ) , where ( x 1 , x 2 , . . . ) ∈ { 0 , 1 } N , and ( y 1 , y 2 , . . . ) ∈ { 0 , 1 } N . The bijective map σ ^ ( . . . , y 2 , y 1 | x 1 , x 2 , . . . ) = ( . . . , y 2 , y 1 , x 1 | x 2 , . . . ) is called the bilateral shift and acts on { 0 , 1 } Z - { 0 } . Given A : { 0 , 1 } N = Ω + → R we express A in the variable x , like A ( x ). In a similar way, given B : { 0 , 1 } N = Ω - → R we express B in the variable y , like B ( y ). Finally, given W : Ω - × Ω + → 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 μ A denotes the equilibrium probability for A : { 0 , 1 } N → R . Given a continuous potential A : Ω + → R , we say that the continuous potential A ∗ : Ω - → R is the dual potential of A , if there exists a continuous W : Ω - × Ω + → R , such that, for all ( y | x ) ∈ { 0 , 1 } Z - { 0 } A ∗ ( y ) = A ∘ σ ^ - 1 + W ∘ σ ^ - 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 : Ω - × Ω + → Ω - × Ω + the function θ ( . . . , y 2 , y 1 | x 1 , x 2 , . . . ) = ( . . . , x 2 , x 1 | y 1 , y 2 , . . . ) . 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 μ A has zero entropy production.