Transient measures in the standard map

For an area preserving map, each chaotic orbit appears numerically to densely cover a region (an irregular component) of nonzero area. Surprisingly, the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component. Most prominently, there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles. We show that this phenomena is transient, and therefore numerical ergodicity on the irregular component eventually obtains, though the times involved are extremely long- 101 ° iterates. A Markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class- for example, a rotational circle. However, the density spikes do occur in a Markov tree model that includes the effects of islands-around-islands. 1. Ergodicity and invariant measures As is well known [1], typical area preserving maps are neither integrable nor ergodic; instead, phase space is divided into a complicated mix-ture of regular and irregular components.