Dimension groups for self-similar maps and matrix representations of the cores of the associated C*-algebras

We introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -group of the core of the C*-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ together with the unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

Simplifying Homeopathy

After the initial enthusiasm in the homeopathic community about the exciting progress and expansion since the 1980s, the sobering fact seems to be that we evolved in the direction of such sophistication that our dear system becomes unmanageable. We did our inner work, try to balance things that were somehow off but still the plain fact is that we are still crushed under too much information. And there is no way to stop it! We are only at the beginning of exploring and including every single species on the planet and beyond in our Materia Medica. But we've already come to a point where even the fastest and most extensive software programs won't help us solve a case. In this article, I argue that we need to boil down the overwhelming bulk of data to clear, simple and reliable pointers to large groups and then smaller groups. The best way to do this is adding the ‘context’ in the analysis. Though often overlooked it turns out to be solid information. The first distinction in a case analysis could be between the Second Dimension (Rocks and Stones, Gems, Bacteria, Viruses, Sarcodes) and the Third Dimension groups (Plants, Animal and Fungi) and often the context will decide. Some information on these groups is given, in an attempt to make homeopathy manageable again without losing its refinement.

Toeplitz flows and their ordered K-theory

To a Toeplitz flow $(X,T)$ we associate an ordered $K^{0}$-group, denoted $K^{0}(X,T)$, which is order isomorphic to the $K^{0}$-group of the associated (non-commutative) $C^{\ast }$-crossed product $C(X)\rtimes _{T}\mathbb{Z}$. However, $K^{0}(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^{0}$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^{+})$ that contain a non-cyclic subgroup $H$ of rank one that intersects $G^{+}$ non-trivially. Furthermore, the Bratteli diagram realization of $(G,G^{+})$ can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex $K$ there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

Real Dimension Groups

Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.