Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita–Takesaki theory in our context.

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Let {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over {\mathbb{K}}. Let {f:X\vdash X} and {g:Y\vdash Y} be dominant correspondences, and {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that {\pi\circ f=g\circ\pi}. We define relative dynamical degrees {\lambda_{p}(f|\pi)\geq 1} for any {p=0,\dots,\dim(X)-\dim(Y)}. These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy {(\varphi,\psi)} from {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when {\mathbb{K}=\mathbb{C}}. Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.