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We obtain a sufficient condition for a substitution ${mathbb Z}$-action to have pure singular spectrum in terms of the top Lyapunov exponent of the spectral cocycle introduced in arXiv:1802.04783 by the authors. It is applied to a family of examples, including those associated with self-similar interval exchange transformations.
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $mathbb{Z}^k$-actions which are generated by random compositions of the generators of $mathbb{Z}^k$-actions. Applying Pesins theory for c
In this paper, directional sequence entropy and directional Kronecker algebra for $mathbb{Z}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum s
It is shown that the Ellis semigroup of a $mathbb Z$-action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asympto
We study directional mean dimension of $mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+infty]$-valued function on the torus)
In this paper, two types of Lyapunov exponents: random Lyapunov exponents and directional Lyapunov exponents, and the corresponding entropies: random entropy and directional entropy, are considered for smooth $mathbb{Z}^k$-actions. The close relation