Computing Invariant Densities and Metric Entropy

We present a method for accurately computing the metric entropy (or, equivalently, the Lyapunov exponent) of the absolutely continuous invariant measure for a piecewise analytic expanding Markov map of the interval. We construct atomic measures M supported on periodic orbits up to period M, and prove that h(M) ! h() super-exponentially fast. We illustrate our method with several examples. 0. The problem Let T : I ! I be a piecewise C 2 expanding map of the interval. By the Lasota-Yorke theorem L-Y] we know there exists a T-invariant probability measure which is absolutely continuous with respect to Lebesgue measure. If T is topologically mixing then this absolutely continuous invariant measure (abbreviated to a.c.i.m.) is unique and ergodic, with strictly positive C 1 density function : I ! R. In the special case of the continued fraction transformation Tx = 1=x (mod 1) the density is (x) = ((1+x) log 2) ?1. For the Ulam-von Neumann map Tx = 4x(1?x), which is topologically conjugate to the expanding tent map (x) = 1 ?j2x?1j, the density is known to be (x) = ?1 (x(1 ? x)) ?1=2. Finally, Parry Pa] and Renyi Renyi] studied the problem for certain piecewise linear examples, including the-transformation deened by Tx = x (mod 1), for any > 1. However, in general there is no explicit formula for the density function, and much recent interest has focused on approximating the density HuntB], Froy1], Froy2], HuntF], K-M-Y], M], with particular emphasis on Ulam's method Ulam]. In this note we will present an alternative approach. We shall only consider the case where T is a real analytic Markov map. In this case a well-known approach to nding the a.c.i.m. is given by the weighted distribution of periodic orbits. More precisely, the sequence of atomic T-invariant probability measures m M = P x2Fix(M) x =j(T M) 0 (x)j P x2Fix(M) 1=j(T M) 0 (x)j ; M 1 (0:1) (where the summations are over the set Fix(M) = fx 2 I : T M x = xg of period-M points) converges to in the weak-star topology (see K-H, p. 635]).