The Ruelle Transfer Operator in the Context of Orthogonal Polynomials

We redefine the Ruelle transfer operator, a classical tool from dynamical systems theory, in terms of orthogonal polynomial sequences. This transfer operator will be given via the preimages of the Chebyshev polynomials of the first kind and we will show that function spaces determined by the Chebyshev polynomials of the first kind are left invariant while function spaces determined by various other orthogonal polynomial sequences are not.