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The multifractal generalizations of Hausdorff dimension and packing dimension are investigated for an invariant subset A of a piecewise monotonic map on the interval. Formulae for the multifractal dimension of an ergodic invariant measure, the essential multifractal dimension of A, and the multifractal Hausdorff dimension of A are derived. Introduction. Consider a piecewise monotonic map T on the interval (exact definitions will be given later), and denote by Z the collection of its intervals of monotonicity. Let U be an open interval and consider the set A of all points x whose orbit {Tnx : n ∈ N0} omits U , that is,

Multifractal dimensions for invariant subsets of piecewise monotonic interval maps