Using tools from computable analysis, we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural systems one can think of are effective in this sense, including some group rotations, affine actions on the torus and finitely presented algebraic actions. We show that for finitely generated and recursively presented groups, every effective dynamical system is the topological factor of a computable action on an effectively closed subset of the Cantor space. We then apply this result to extend the simulation results available in the literature beyond zero-dimensional spaces. In particular, we show that for a large class of groups, many of these natural actions are topological factors of subshifts of finite type.

Two asymptotic configurations on a full $\mathbb {Z}^d$-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb {Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $\mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.

We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed $\mathbb{Z}^{d}$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^{2}$. Let $H$ be a finitely generated group and $G=\mathbb{Z}^{2}\rtimes _{\unicode[STIX]{x1D711}}H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,T)$ where $Y\subset \{0,1\}^{\mathbb{N}}$, there exists a $G$-subshift of finite type $(X,\unicode[STIX]{x1D70E})$ such that the $H$-subaction of $(X,\unicode[STIX]{x1D70E})$ is an extension of $(Y,T)$. In the case where $T$ is an expansive action, a subshift conjugated to $(Y,T)$ can be obtained as the $H$-projective subdynamics of a sofic $G$-subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.