The boundary quotient for algebraic dynamical systems

A Fej\'{e}r-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system ( f , X , Ω ) (f,X,\Omega ) . The main idea is to make use of, in addition to invariant compact subspaces of ( X , Ω ) (X,\Omega ) , compactifications of étale covers π : ( f ′ , X ′ , Ω ′ ) → ( f , X , Ω ) \pi :(f’,X’,\Omega ’)\rightarrow (f,X,\Omega ) ; that is, π ∘ f ′ = f ∘ π \pi \circ f’=f\circ \pi and the fibers of π \pi are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an étale analog of algebraic dynamical systems.

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.

The equidistribution conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan’s equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov conjecture and hence a self-contained proof of the strong equidistribution conjecture in the same general setting.

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.

Traces of algebraic integers and dynamical systems

Given a matroid M on the ground set E, the Bergman fan e B(M), or space of M- ultrametrics, is a polyhedral complex in R E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situ- ation, we study the following problem: Given a point ω in R E , we wish to nd an M-ultrametric which is closest to it in the '¥-metric. The solution to this problem follows easily from the existence of the subdominant M-ultramet- ric: a componentwise maximum M-ultrametric which is componentwise smaller than ω. A pro- cedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry. When the matroid in question is the graphical matroid of the complete graph Kn, the Bergman fan e B(Kn) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.

C*-algebraic partial dynamical systems

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials $$\{x^N\}_{N=0}^\infty $$ . We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.

Multicyclic Codes and Algebraic Dynamical Systems

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

We study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahler measure of the first non-zero Alexander polynomial of the corresponding modules. We use the notion of pseudo-isomorphism, and also tools from commutative algebra and algebraic geometry, to reduce the conjectures to the case of torsion modules. We also describe concrete sequences which give the expected values of the limits in both cases. For this part we utilize a result of Bombieri and Zannier (conjectured before by A. Schinzel) and a result of Lawton (conjectured before by D. Boyd).