We address the problem of the multifractal analysis of local entropies for arbitrary invariant measures. We obtain an upper estimate on the multifractal spectrum of local entropies, which is similar to the estimate for local dimensions. We show that in the case of Gibbs measures the above estimate becomes an exact equality. In this case the multifractal spectrum of local entropies is a smooth concave function. We discuss possible singularities in the multifractal spectrum and their relation to phase...

We introduce the notion of a generalized interval exchange $${\phi }_{𝒜}$$ induced by a measurable k-partition $$𝒜=\left\{A_1,...,A_k\right\}$$ of [0,1). $${\phi }_{𝒜}$$ can be viewed as the corresponding restriction of a nondecreasing function $$f_{𝒜}$$ on ℝ with $$f_{𝒜}\left(0\right)=0,f_{𝒜}\left(k\right)=1$$ . A is called λ-dense if λ(A i∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that $$f_{𝒜}\circ f_{ℬ}=f_{ℬ}\circ f_{𝒜}$$ . We give necessary and sufficient conditions for this equality to hold. We show that...