typically be very closely spaced, and thus even a very narrow and carefully selected ensemble of quantum states will encompass a large number of very different fractal eigenstates with non-zero support over the entire volume. This can be used to explain many results from classical statistical mechanics, including the irreversibility of time and the increase of entropy. The transfer operator of the Bernoulli map is exactly solvable and is a classic example of deterministic chaos; the discrete eigenvalues correspond to the Bernoulli polynomials. This operator also has a continuous spectrum consisting of the Hurwitz zeta function. The transfer operator of the Gauss map is called the Gauss–Kuzmin–Wirsing (GKW) operator. The theory of the GKW dates back to a hypothesis by Gauss on continued fractions and is closely related to the Riemann zeta function. See also Bernoulli scheme Shift of finite type Krein–Rutman theorem References (Provides an introductory survey). Chaos theory Dynamical systems Operator theory Spectral theory