We study the Besicovitch pseudometric D B for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to D B . It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure µ of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for µ. A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.

We study the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both lower semicontinuous with respect to the Weyl pseudometric. Furthermore, the simplex of invariant measures supported on the orbit closure varies continuously. We apply the Weyl pseudometric to Toeplitz configurations for arbitrary amenable residually finite groups. We introduce the notion of a regular Toeplitz configuration and demonstrate that all regular Toeplitz configurations define minimal and uniquely ergodic systems. We prove that this family is path-connected in the Weyl pseudometric. This leads to a new proof of a theorem of Krieger, saying that Toeplitz configurations can have arbitrary finite entropy.is ergodic if for every G-invariant set A ⊂ X one has either μ(A) = 0 or μ(A) = 1. We denote by † We omit this assumption at the beginning of Section 7. ‡ We used the adjective to distinguish left Følner sequences from their right-and two-sided analogues. In a similar vain notions related to non-commutative groups have 'left' and 'right' variants. For brevity, we will use adjectives 'left/right' only in definitions and routinely omit them later, since the choice of 'left/right' is fixed throughout the paper.

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The aim of this note is to give an alternative proof for the following result originally proved by Bonatti, Díaz and Kwietniak. For every $$n\ge 3$$ n ≥ 3 there exists a compact manifold without boundary $${\mathbf {M}}$$ M of dimension n and a non-empty open set $$U\subset \text {Diff}({\mathbf {M}})$$ U ⊂ Diff ( M ) such that for every $$f\in U$$ f ∈ U there exists a non-hyperbolic measure $$\mu $$ μ invariant for f with positive entropy and full support. We also investigate the connection between the Feldman-Katok convergence of measures and the Kuratowski convergence of their supports.

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