Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent

We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains

Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices

We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.Comment: 18 page

Performance Assessment and English Skill.

Recent developments in the philosophy of validity, highlighting the importance of investigating the consequences of assessment use, provide theoretical support for the move toward performance assessment. Numerous issues related to the choice of assessment approach (e.g., multiple-choice or performance-based) are not well understood today. This study investigated the potential interaction between assessment approach and the English skill of the examinee and the similarity of the information provided by multiple-choice and essay writing tests. Four hypotheses were examined in this study. Hypothesis 1 results indicated that students who were native speakers of English (regular students) achieved higher scores on the writing test than did the LEP students. Regular students were significantly different from LEP students in support/elaboration/organizations, sentence structure, and usage. However, there were no significant differences between these groups in writing mechanics and responsiveness to assignment. The results for Hypothesis 2 indicated that regular students scored significantly higher in all the eight subskills of the multiple-choice test than did the LEP students. Hypothesis 3 indicated a substantial interaction between English skill and assessment approach. The differences between the two groups on the higher-order elements of composition were much greater on the writing test than the multiple-choice test. Finally, the results for hypothesis 4 indicated that the factorial structure of the two tests were essentially identical for the two groups. It was concluded that the multiple-choice test consistently differentiated between the two groups of students studied in the direction consistent with the English skill levels of the students. In contrast, the performance-based writing test yielded expected differences for higher-order skills, but failed to separate the groups on lower level writing skills. This suggests that the multiple-choice test is more general in its utility than is true of the performance-based writing test. Also, it suggests that the writing test may be superior to the multiple-choice test at differentiating students on the higher-order elements of compositions. Conversely, it may be the case that the writing test exaggerates the differences between the groups raising the possibility of bias. The implications of this for high-stakes testing programs are discussed and recommendations are offered