Uniqueness of spaces pretangent to metric spaces at infinity

We find the necessary and sufficient conditions under which an unbounded metric space \(X\) has, at infinity, a unique pretangent space \(\Omega^{X}_{\infty,\tilde{r}}\) for every scaling sequence \(\tilde{r}\). In particular, it is proved that \(\Omega^{X}_{\infty,\tilde{r}}\) is unique and isometric to the closure of \(X\) for every logarithmic spiral \(X\) and every \(\tilde{r}\). It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the ''asymptotic asymmetry'' of these subsets.

COMPACTNESS OF CERTAIN FAMILIES OF PSEUDO-HOLOMORPHIC MAPPINGS INTO ${\mathbb C}^n$

We present a normal family theorem for injective almost holomorphic maps from a manifold with almost complex structures into [Formula: see text]. Our theorem implies a new consequence even for the holomorphic mappings of a complex manifold into [Formula: see text], which can be seen as a generalization of the convergence theorem for Frankel's scaling sequence whose images are not necessarily convex. Moreover, our method is closer in spirit to the circle of ideas centered around the classical Montel theorem.

SELF-SIMILAR STRUCTURE OF RESCALED EVOLUTION SETS OF CELLULAR AUTOMATA II

The description of the rescaled evolution set of p-Fermat cellular automata developed in the first part of this paper is applied for the calculation of the Hausdorff dimension of this set. The scaling procedure is analyzed and an appropriate scaling sequence for cellular automata with states in the integers modulo m is given. New proof of the main theorem of the first part of the paper is presented for the linear cellular automata with states in the integers modulo a power of prime number.