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.

UNDERSTANDING CUSTOMER CHOICES IN SERVICE OUTSOURCING AND REVENUE MANAGEMENT

This dissertation investigates customer behavior modeling in service outsourcing and revenue management in the service sector (i.e., airline and hotel industries). In particular, it focuses on a common theme of improving firms’ strategic decisions through the understanding of customer preferences. Decisions concerning degrees of outsourcing, such as firms’ capacity choices, are important to performance outcomes. These choices are especially important in high-customer-contact services (e.g., airline industry) because of the characteristics of services: simultaneity of consumption and production, and intangibility and perishability of the offering. Essay 1 estimates how outsourcing affects customer choices and market share in the airline industry, and consequently the revenue implications from outsourcing. However, outsourcing decisions are typically endogenous. A firm may choose whether to outsource or not based on what a firm expects to be the best outcome. Essay 2 contributes to the literature by proposing a structural model which could capture a firm’s profit-maximizing decision-making behavior in a market. This makes possible the prediction of consequences (i.e., performance outcomes) of future strategic moves. Another emerging area in service operations management is revenue management. Choice-based revenue systems incorporate discrete choice models into traditional revenue management algorithms. To successfully implement a choice-based revenue system, it is necessary to estimate customer preferences as a valid input to optimization algorithms. The third essay investigates how to estimate customer preferences when part of the market is consistently unobserved. This issue is especially prominent in choice-based revenue management systems. Normally a firm only has its own observed purchases, while those customers who purchase from competitors or do not make purchases are unobserved. Most current estimation procedures depend on unrealistic assumptions about customer arriving. This study proposes a new estimation methodology, which does not require any prior knowledge about the customer arrival process and allows for arbitrary demand distributions. Compared with previous methods, this model performs superior when the true demand is highly variable.