In a recent article D. Ruelle [in Lecture Notes in Physics, No. 80 (Springer, Berlin, 1978)] has conjectured that for the H~non attractor its measure theoretic entropy should be equal to its characteristic exponent. This result is known to be true for systems which satisfy Smale's Axiom A. In this article we report the results of our computations which suggest that Ruelle's conjecture may be true for the H~non attractor. Further, in our study we are confronted'with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view.