A metric ρ_I that defines a metrization of the functor I of idempotent measures is introduced on the space I(X) of idempotent measures (Maslov measures) given on a compact metric space X . For each measure μ∈ I(X) , the upper and lower quantization dimensions of this measure can be defined with respect to the metric ρ_I ; they do not exceed the corresponding box dimensions of the support of the measure μ . The following theorem on intermediate values of quantization dimensions of idempotent measures is proved: on every compact metric space of box dimension a<∞ , for all two reals b and c related by the inequalities 0≤ b≤ c≤ a , there exists an idempotent measure whose lower and upper quantization dimensions are equal to b and c , respectively. An analogous theorem on intermediate values was previously proved by the author for quantization dimensions of probability measures, whose idempotent analogs are Maslov measures.
We describe derivations of finite-dimensional simple Novikov algebras and pre-Lie Witt doubles over an algebraically closed field of positive characteristic in terms of special derivations of the underlying associative commutative differentially simple algebras.
