Bifurcations in Complex Algebraic Dynamics

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The stability of a dynamical system is arguably its most important feature, from a theoretical, computational, or practical point of view. For systems that evolve with time, one aims to determine which perturbations will preserve the system’s long-term behavior and which perturbations will lead to radically different outcomes. This project concerns stability and bifurcations in the setting of complex algebraic dynamical systems. Such systems are defined by polynomial formulas in one or several variables. The algebraic nature of the defining equations connect the dynamical study with the rich theory of algebraic geometry. Moreover, in the case of examples where all of the defining polynomials have, for example, integer coefficients, the relevant dynamical stability questions have surprising connections to number theory and to the Diophantine geometry of the underlying equations. The project will extend the theory of dynamical stability for complex analytic examples to new settings that arise naturally in arithmetic geometry and complex dynamics. The project also provides research and training opportunities for graduate students and postdocs.This project develops the theory of stability for analytic families of maps on projective spaces, in both a complex analytic setting and in the setting of non-archimedean-valued fields and p-adic analysis. It was recently discovered, in earlier work of the PI and of other researchers, that certain questions about height functions and arithmetic intersection theory can be analyzed using complex dynamics. In a series of recent breakthroughs in arithmetic geometry, especially concerning uniform bounds for numbers of rational points on families of algebraic varieties, stability theory played a crucial--if somewhat hidden--role. This project aims to shed new light on the role of stability theory and to push the theory further. Many of the proposed problems and applications of the theory are related to the occurrence of `unlikely intersections’ in families of abelian varieties or in more general families of polarized dynamical systems. Specific goals of this project include (1) to characterize positivity properties of certain bifurcation currents and measures; (2) to provide bounds on the geometry of invariant subvarieties for algebraic dynamical systems; and (3) to formulate a theory of bifurcations in the setting of p-adic analytic families of maps. The research activity conducted under this award is expected to impact multiple areas of mathematics, including number theory, geometry, and dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.