Regular homomorphisms of minimal sets

The classification of minimal sets is a central theme in abstract topological dynamics. Recently this work has been strengthened and extended by consideration of homomorphisms. Background material is presented in Chapter I. Given a flow on a compact Hausdorff space, the action extends naturally to the space of closed subsets, taken with the Hausdorff topology. These hyperspaces are discussed and used to give a new characterization of almost periodic homomorphisms. Regular minimal sets may be described as minimal subsets of enveloping semigroups. Regular homomorphisms are defined in Chapter II by extending this notion to homomorphisms with minimal range. Several characterizations are obtained. In Chapter III, some additional results on homomorphisms are obtained by relativizing enveloping semigroup notions. In Veech's paper on point distal flows, hyperspaces are used to associate an almost one-to-one homomorphism with a given homomorphism of metric minimal sets. In Chapter IV, a non-metric generalization of this construction is studied in detail using the new notion of a highly proximal homomorphism. An abstract characterization is obtained, involving only the abstract properties of homomorphisms. A strengthened version of the Veech Structure Theorem for point distal flows is proved. In Chapter V, the work in the earlier chapters is applied to the study of homomorphisms for which the almost periodic elements of the associated hyperspace are all finite. In the metric case, this is equivalent to having at least one fiber finite. Strong results are obtained by first assuming regularity, and then assuming that the relative proximal relation is closed as well.

Ionospheric Turbulence Near the Upper Hybrid Layer: Theory and Experiment

The thesis presents experimental results, simulations, and theory on turbulence excited in magnetized plasmas near the ionosphere’s upper hybrid layer. The results include: The first experimental observations of super small striations (SSS) excited by the High-Frequency Auroral Research Project (HAARP) The first detection of high-frequency (HF) waves from the HAARP transmitter over a distance of 16x10^3 km The first simulations indicating that upper hybrid (UH) turbulence excites electron Bernstein waves associated with all nearby gyroharmonics Simulation results that indicate that the resulting bulk electron heating near the upper hybrid (UH) resonance is caused primarily by electron Bernstein waves parametrically excited near the first gyroharmonic. On the experimental side we present two sets of experiments performed at the HAARP heating facility in Alaska. In the first set of experiments, we present the first detection of super-small (cm scale) striations (SSS) at the HAARP facility. We detected density structures smaller than 30 cm for the first time through a combination of satellite and ground based measurements. In the second set of experiments, we present the results of a novel diagnostic implemented by the Ukrainian Antarctic Station (UAS) in Verdansky. The technique allowed the detection of the HAARP signal at a distance of nearly 16 Mm, and established that the HAARP signal was injected into the ionospheric waveguide by direct scattering off of dekameter-scale density structures induced by the heater. On the theoretical side, we present results of Vlasov simulations near the upper hybrid layer. These results are consistent with the bulk heating required by previous work on the theory of the formation of descending artificial ionospheric layers (DIALs), and with the new observations of DIALs at HAARP’s upgraded effective radiated power (ERP). The simulations that frequency sweeps, and demonstrate that the heating changes from a bulk heating between gyroharmonics, to a tail acceleration as the pump frequency is swept through the fourth gyroharmonic. These simulations are in good agreement with experiments. We also incorporate test particle simulations that isolate the effects of specific wave modes on heating, and we find important contributions from both electron Bernstein waves and upper hybrid waves, the former of which have not yet been detected by experiments, and have not been previously explored as a driver of heating. In presenting these results, we analyzed data from HAARP diagnostics and assisted in planning the second round of experiments. We integrated the data into a picture of experiments that demonstrated the detection of SSS, hysteresis effects in simulated electromagnetic emission (SEE) features, and the direct scattering of the HF pump into the ionospheric waveguide. We performed simulations and analyzed simulation data to build the understanding of collisionless heating near the upper hybrid layer, and we used these simulations to show that bulk electron heating at the upper hybrid layer is possible, which is required by current theories of DAIL formation. We wrote a test particle simulation to isolate the effects of electron Bernstein waves and upper hybrid layers on collisionless heating, and integrated this code to work with both the output of Vlasov simulations and the input for simulations of DAIL formation.

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.

Trade Shocks and Local Labor Markets' Linkages: Theory and Evidence

In the past few years, there has been a concern among economists and policy makers that increased openness to international trade affects some regions in a country more than others. Recent research has found that local labor markets more exposed to import competition through their initial employment composition experience worse outcomes in several dimensions such as, employment, wages, and poverty. Although there is evidence that regions within a country exhibit variation in the intensity with which they trade with each other and with other countries, trade linkages have been ignored in empirical analyses of the regional effects of trade, which focus on differences in employment composition. In this dissertation, I investigate how local labor markets' trade linkages shape the response of wages to international trade shocks. In the second chapter, I lay out a standard multi-sector general equilibrium model of trade, where domestic regions trade with each other and with the rest of the world. Using this benchmark, I decompose a region's wage change resulting from a national import cost shock into a direct effect on prices, holding other endogenous variables constant, and a series of general equilibrium effects. I argue the direct effect provides a natural measure of exposure to import competition within the model since it summarizes the effect of the shock on a region's wage as a function of initial conditions given by its trade linkages. I call my proposed measure linkage exposure while I refer to the measures used in previous studies as employment exposure. My theoretical analysis also shows that the assumptions previous studies make on trade linkages are not consistent with the standard trade model. In the third chapter, I calibrate the model to the Brazilian economy in 1991--at the beginning of a period of trade liberalization--to perform a series of experiments. In each of them, I reduce the Brazilian import cost by 1 percent in a single sector and I calculate how much of the cross-regional variation in counterfactual wage changes is explained by exposure measures. Over this set of experiments, employment exposure explains, for the median sector, 2 percent of the variation in counterfactual wage changes while linkage exposure explains 44 percent. In addition, I propose an estimation strategy that incorporates trade linkages in the analysis of the effects of trade on observed wages. In the model, changes in wages are completely determined by changes in market access, an endogenous variable that summarizes the real demand faced by a region. I show that a linkage measure of exposure is a valid instrument for changes in market access within Brazil. By using observed wage changes in Brazil between 1991-2000, my estimates imply that a region at the 25th percentile of the change in domestic market access induced by trade liberalization, experiences a 0.6 log points larger wage decline (or smaller wage increase) than a region at the 75th percentile. The estimates from a regression of wages changes on exposure imply that a region at the 25th percentile of exposure experiences a 3 log points larger wage decline (or smaller wage increase) than a region at the 75th percentile. I conclude that estimates based on exposure overstate the negative impact of trade liberalization on wages in Brazil. In the fourth chapter, I extend the standard model to allow for two types of workers according to their education levels: skilled and unskilled. I show that there is substantial variation across Brazilian regions in the skill premium. I use the exogenous variation provided by tariff changes to estimate the impact of market access on the skill premium. I find that decreased domestic market access resulting from trade liberalization resulted in a higher skill premium. I propose a mechanism to explain this result: that the manufacturing sector is relatively more intensive in unskilled labor and I show empirical evidence that supports this hypothesis.