The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank-$1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$, and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

We generalize the influential $C^*$-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\,\infty ]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of $\omega $-categorical structures on which the Zariski topology has been analysed thus far, the two topologies were shown to coincide, in turn yielding that the pointwise topology is the coarsest Hausdorff semigroup topology on those endomorphism monoids.

We establish two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, we give an example of an $\omega $-categorical structure on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonušas, Mitchell, Péresse, and Pinsker.

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.

As a by-product, we show some analogous results in purely topological context (without direct use of model theory).

We extend the classical notion of an outer action $\alpha $ of a group $G$ on a unital ring $A$ to the case when $\alpha $ is a partial action on ideals, all of which have local units. We show that if $\alpha $ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A\,{{\star }_{\alpha }}\,G$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

We prove that a connected, countable dense homogeneous space is $n$-homogeneous for every n, and strongly 2-homogeneous provided it is locally connected. We also present an example of a connected and countable dense homogeneous space which is not strongly 2-homogeneous. This answers in the negative Problem 136 ofWatson in the Open Problems in Topology Book.

If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\text{Spec}\,H_{T}^{*}\left( X;\,\mathbb{C} \right)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_{2}^{T}\left( X;\,\mathbb{C} \right)$. In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.