A construction of singular overlapping asymmetric self-similar measures (Q879212)

The study of Bernoulli measures, i.e. of symmetric self similar measures has a long history, reported in the paper of \textit{Y. Peres, W. Schlag} and \textit{B. Solomyak} [Prog. Probab. 46, 39--65 (2000; Zbl 0961.42006)]. The author started a generalization of these measures by considering overlapping asymmetric self-similar measures on the real line, being defined by two parameters, see \textit{J. Neunhäuserer} [Acta. Math. Hungar. 93, 143--161 (2001; Zbl 1002.28011)]. As a result he found, that overlapping asymmetric selfsimilar measures are generically absolutely continuous with respect to the Lebesgue measure on the real line. The present article deals with the question, whether there exist exceptional overlapping asymmetric self-similar measures being singular. This question is answered to the positive by showing that near the boundary of the parameter domain there exist exceptional values for which the overlapping asymmetric self-similar measures get singular. More precisely it is shown that each point of this boundary is an accumulation point of parameters for which the (Hausdorff) dimension of the measures is less than one, hence singular. The result is based on a dimension estimate, in which the dimension is bounded by quotient of entropy and Lyapunov exponent. The latter can be computed. This estimate uses first classical proceedures from ergodic theory and in particular it applies a refinement of a construction given by \textit{K. Simon} and \textit{B. Solomyak} in [Fractals 10, 59--65 (2002; Zbl 1088.28504)].