ИСТИНА

Multivariate wavelets on~${\mathbf R}^d$ can be constructed with an arbitrary dilation integer expansive $d\times d$ matrix and arbitrary set of ``digits'' from the corresponding quotient sets. A formula for the H\"older exponent of multivariate wavelets constructed with an arbitrary dilation matrix was presented M.Charina, V.Yu.Protasov, Regularity of anisotropic refinable functions, ACHA (2017), https://doi.org/10.1016/j.acha. We show that a similar techniques can be used to make the high order regularity analysis, in particular, for the multivariate B-splines. In the simplest case of multivariate Haar wavelets, this leads to computing the boundary dimension of self-similar tiles. Moreover, the same value has an interpretation in terms of the problem of synchronizing automata. A finite automata is determined by a directed multigraph with $N$ vertices (states) and with all edges (transfers) coloured with $m$ colours so that each vertex has precisely one outgoing edge of each colour. The automata is synchronizing if there exists a finite sequence of colours such that all paths following that sequence terminate at the same vertex independently of the starting vertex. The problem of synchronizing automata has been studied in great detail. It turns out that each multivariate Haar function can be naturally associated with a finite automata and the H\"older exponent is related to the length of the synchronizing sequence. We introduce a concept of synchronizing rate and show that it is actually equal to the H\"older exponent of the corresponding Haar function. Applying this result we prove that the boundary dimension of self-similar tiles, as well as the H\"older exponent of Haar functions, can be found within finite time by a combinatorial algorithm.