Polynomial Shift States of a Chaotic Map

We present a piecewise- linear map of the unit interval in which the resolvent of the Frobenius- Perron operator, considered in a polynomial basis, has an essen-tial singularity at the origin. Associated with the essential singularity are polynomial shift states, which are obtained from creation and annihilation operators in non- self- dual function spaces. Correlation functions of general polynomial observables have decay components that vanish in a finite time.