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We characterize ballistic behavior for general i.i.d. random walks in random environments on Z with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. MSC 2020itions. : 60G50

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show they are closed under composition. We also study a family of n-ary interleaving operations, one for each n ≥ 1. Given subsets X0, X1, ..., Xn−1 of the shift space, the n-ary interleaving operator produces a set whose elements combine individual elements xi, one from each Xi, by interleaving their symbol sequences cyclically in arithmetic progressions (mod n). We determine algebraic relations between decimation and interleaving operators and the shift operator. We study set-theoretic n-fold closure operations X → X [n] , which interleave decimations of X of modulus level n. A set is n-factorizable if X = X [n] . The n-fold interleaving operators are closed under composition and are idempotent. To each X we assign the set N (X) of all values n ≥ 1 for which X = X [n] . We characterize the possible sets N (X) as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. This class includes all shift-invariant sets. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable X, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

For a shift-invariant weighted directed graph with vertex set Z, we examine the minimal weight κ 0 exiting a finite, strongly connected set of vertices. Although κ 0 is defined as an infimum, it has been shown that the infimum is always attained by an actual set of vertices. We show that for each underlying directed graph (prior to assignment of the weights), there is a formula for κ 0 as a minimum of finitely many integer combinations of the edge weights. We find this formula for several different directed graphs. Motivation for this problem comes from random walks in Dirichlet environments (equivalently, directed edge reinforced random walks), where the size of κ 0 has been shown to determine the strength of finite traps where the walk can get stuck for a long time.