Critical lattices, elliptic curves and their possible dynamics (original) (raw)
Dynamical systems arising from elliptic curves
Colloquium Mathematicum, 2000
We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose entropy is the global canonical height and whose periodic points are counted asymptotically by the real division polynomial (although the archimedean component of the map is artificial). Finally, we set out a precise conjecture about the existence of elliptic dynamical systems and discuss a possible connection with mathematical physics.
A Dynamical Interpretation of the Global Canonical Height on an Elliptic Curve
Experimental Mathematics, 1998
There is a well-understood connection between polynomials and certain simple algebraic dynamical systems. In this connection, the Mahler measure corresponds to the topological entropy, Kronecker's Theorem relates ergodicity to positivity of entropy, approximants to the Mahler measure are related to growth rates of periodic points, and Lehmer's problem is related to the existence of algebraic models for Bernoulli shifts. There are similar relationships for higher-dimensional algebraic dynamical systems. We review this connection, and indicate a possible analogous connection between the global canonical height attached to points on elliptic curves and a possible 'elliptic' dynamical system.
Some applications of homogeneous dynamics to number theory
2002
This survey paper is not a complete reference guide to number-theoretical applications of ergodic theory. Instead, the plan is to consider an approach to a class of problems involving Diophantine properties of n-tuples of real numbers, namely, describe a specific dynamical system which is naturally connected with these problems.
Trends in Mathematics, 2017
Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form Q(z) dz 2 = az 2 +bz+c (z 2 -1) 2 dz 2 . Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.
Borcea's variance conjectures on the critical
2016
Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures.