Entropy and growth rate of periodic points of algebraic Z^d-actions

Expansive algebraic Z^d-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable.Comment: 17 page

Non-archimedean amoebas and tropical varieties

We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.Comment: 19 pages, 2 figures. Added AIM preprint numbe

Doctrines of Discovery

The idea that “discovery” of unknown lands carried with it the right to assert sovereignty and claim ownership was widely used by European sovereigns during the age of modern colonialization to justify appropriating indigenous lands. Felix Cohen’s pioneering work in the 1940s on federal Indian law made discovery a matter of jurisprudential interest and highlighted its role in advancing the English colonial empire in what became the United States. Specifically, Cohen argued that the natural law right of discovery, as formulated by Spanish philosopher Francisco de Vitoria, helped facilitate the early European settlement of the American colonies and became a bedrock of federal Indian law. Today, legal scholars in the United States and elsewhere across the former British Empire view discovery as a discredited idea that contributed painfully to the displacement of indigenous peoples. That scholarship is incisive and valuable. Yet it contains a characteristic feature which exposes a serious flaw in Cohen’s work. The characteristic feature is the treatment of discovery as an idea manifested in a single “Doctrine of Discovery” purportedly accepted as a principle of international law influencing European adventurism beginning in Iberia during the Renaissance and continuing throughout the colonial era. The serious flaw is that this single doctrine of discovery thesis originated with Cohen and is mistaken. In this paper, I question Cohen’s claims about the influence of Vitoria’s right of discovery on United States federal Indian law. More generally, I question the thesis that a single doctrine of discovery held sway for centuries in international law guiding European exploration and appropriation of indigenous lands. I argue that the history of jurisprudential thought and legal decision does not support the single discovery doctrine thesis. Rather, the idea of discovery appeared in a number of distinct theories favored by European powers in different ages and geopolitical contexts. I identify and distinguish four different discovery doctrines: (1) the medieval papal theory of discovery which helped spread Christianity across Europe and beyond beginning in the Middle Ages; (2) the natural law right of discovery begun by Vitoria in the 1530s and refined by later philosophers writing in the traditions of natural law and the law of nations; (3) the form the idea of discovery took with the United States Supreme Court early in the nineteenth century; and (4) the discovery theory of terra nullius employed by the British in settling Australia. I conclude that carefully distinguishing the different ideas of discovery is necessary to address and seek recompense for specific instances of indigenous dispossession and displacement