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We show that for almost every map in a transversal one-parameter family ofpiecewise expanding unimodal maps the Birkhoff sum of suitable observablesalong the forward orbit of the turning point satisfies the law of iteratedlogarithm. This result will follow from an almost sure invariance principle forthe Birkhoff sum, as a function on the parameter space. Furthermore, we obtaina similar result for general one-parameter families of piecewise expanding mapson the interval.

Law of iterated logarithm and invariance principle for one-parameter families of interval maps