We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕn. If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ.

To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.

We consider a family of discrete Jacobi operators on the one-dimensional integer latticewith Laplacian and potential terms modulated by a primitive invertible two-lettersubstitution. We investigate the spectrum and the spectral type, the fractal structure andfractal dimensions of the spectrum, exact dimensionality of the integrated density ofstates, and the gap structure. We present a review of previous results, some applications,and open problems. Our investigation is based largely on the dynamics of trace maps. Thiswork is an extension of similar results on Schrödinger operators, although some of theresults that we obtain differ qualitatively and quantitatively from those for theSchrödinger operators. The nontrivialities of this extension lie in the dynamics of theassociated trace map as one attempts to extend the trace map formalism from theSchrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are,in a sense, a test item, as many other models can be attacked via the same techniques, andwe present an extensive discussion on this.