Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

The role of the Indian Ocean sector and sea surface salinity for prediction of the coupled Indo-Pacific system

The purpose of this dissertation is to evaluate the potential downstream influence of the Indian Ocean (IO) on El Niño/Southern Oscillation (ENSO) forecasts through the oceanic pathway of the Indonesian Throughflow (ITF), atmospheric teleconnections between the IO and Pacific, and assimilation of IO observations. Also the impact of sea surface salinity (SSS) in the Indo-Pacific region is assessed to try to address known problems with operational coupled model precipitation forecasts. The ITF normally drains warm fresh water from the Pacific reducing the mixed layer depths (MLD). A shallower MLD amplifies large-scale oceanic Kelvin/Rossby waves thus giving ~10% larger response and more realistic ENSO sea surface temperature (SST) variability compared to observed when the ITF is open. In order to isolate the impact of the IO sector atmospheric teleconnections to ENSO, experiments are contrasted that selectively couple/decouple the interannual forcing in the IO. The interannual variability of IO SST forcing is responsible for 3 month lagged widespread downwelling in the Pacific, assisted by off-equatorial curl, leading to warmer NINO3 SST anomaly and improved ENSO validation (significant from 3-9 months). Isolating the impact of observations in the IO sector using regional assimilation identifies large-scale warming in the IO that acts to intensify the easterlies of the Walker circulation and increases pervasive upwelling across the Pacific, cooling the eastern Pacific, and improving ENSO validation (r ~ 0.05, RMS~0.08C). Lastly, the positive impact of more accurate fresh water forcing is demonstrated to address inadequate precipitation forecasts in operational coupled models. Aquarius SSS assimilation improves the mixed layer density and enhances mixing, setting off upwelling that eventually cools the eastern Pacific after 6 months, counteracting the pervasive warming of most coupled models and significantly improving ENSO validation from 5-11 months. In summary, the ITF oceanic pathway, the atmospheric teleconnection, the impact of observations in the IO, and improved Indo-Pacific SSS are all responsible for ENSO forecast improvements, and so each aspect of this study contributes to a better overall understanding of ENSO. Therefore, the upstream influence of the IO should be thought of as integral to the functioning of ENSO phenomenon.