Abstract
We introduce the notion of a “weakly attracting repellor” in one dimensional dynamical systems, which is an unstable fixed point with the following property: the time average of ‘distance’ between the fixed point and the iterates of a pointx converges to 0 for almost every pointx. We study two classes of piecewise convex maps. One is a class of piecewise convex increasing maps including some intermittent dynamical systems, and the other is a class of cusp type maps which is related to the Lorenz equation with a critical parameter. We give some conditions for a piecewise convex map having a weakly attracting repellor. We also study the Lyapunov numbers of piecewise convex maps with weakly attracting repellors. The key to our proof is a δ-finite invariant measure.
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