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Misiurewicz maps are rare. (English) Zbl 0921.58015
An \(S\)-unimodal map with a non-recurrent critical point and without periodic attractors is known as a Misiurewicz map. Misiurewicz asked whether in one-parameter families like the logistic one, i.e., \[ x\mapsto 4ax(1-x), \quad 0<a\leq 4, \] the set of parameter values has Lebesgue measure zero or not. The main result of this paper is that the set of Misiurewicz parameters has Lebesgue measure zero in any real analytic family of \(S\)-unimodal maps with non-constant combinatorics. In particular, this is the case for the logistic family, thereby answering an old question.

37E99 Low-dimensional dynamical systems
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