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Nguyen-Bac Dang

Publications and source records attributed to Nguyen-Bac Dang.

12 recordsLinked to original sources
arXiv2026

Equidistribution of currents under Anosov group actions

We study the behavior of currents on flag-manifolds under actions of Anosov subgroups of complex semisimple Lie groups G. Given a current T of bidimension (k, k) on the full flag-manifold F = G/B, we average it under the group action via a construction analogous to the construction of Patterson-Sullivan measures. We show that under certain genericity conditions (in the case of currents of integration over subvarieties), the limiting current is a Gibbs current, i.e. is given by integration (with respect to a Gibbs measure) over the flag-limit set of suitable multiples of currents of integration along Schubert varieties based at the limit points of the group. We prove that the same equidistribution result (but without any genericity assumptions) for currents defined by smooth forms on F. We also prove a form of Axiom A property for the suspension flow of the group action on F.

math.DS↗
arXiv2024

Variation of the Hausdorff dimension and degenerations of Schottky groups

We show that the Hausdorff dimension of the limit set of a Schottky group varies continuously over the moduli space of Schottky groups defined over any complete valued field constructed by Poineau and Turchetti. To obtain this result, we first study the non-Archimedean case in a setting of Berkovich analytic spaces, where we make use of Poincaré series. We show that the latter can be extended meromorphically over the complex plane and admit a special value at zero which is a purely topological invariant. As an application, we prove results on the asymptotic behavior of the Hausdorff dimension of degenerating families of complex Schottky groups. For certain families, including Schottky reflection groups, we obtain an exact formula for the asymptotic logarithmic decay rate of the Hausdorff dimension. This generalizes a theorem of McMullen.

math.AG↗
arXiv2021

Intersection theory of nef b-divisor classes

We prove that any nef b-divisor class on a projective variety defined over an algebraically closed field of characteristic 0 is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to Zariski decompositions of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of b-divisors which were defined in our previous work.5

math.AG↗
arXiv2021

Spectral interpretations of dynamical degrees and applications

We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space of b-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimensions of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in this Banach space. We use these two theorems to infer a precise control of the sequence of degrees of iterates of a map under the assumption that the square of the first dynamical degree is strictly larger than the second dynamical degree. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine 3-space are all algebraic numbers.

math.AG↗
arXiv2021

Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of discrete spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

math.GR↗
arXiv2020

Dynamical degrees of automorphisms on abelian varieties

For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.

math.AG↗
arXiv2020

Dynamical invariants of monomial correspondences

We focus on various dynamical invariants associated to toric correspondences, using algebraic geometry or arithmetic. We find a formula for the dynamical degrees, relate the exponential growth of the degree sequences with a strict log-concavity condition on the dynamical degrees and compute the asymptotic ratio of the growth of heights of points of such correspondences.

math.DS↗
arXiv2019

Degrees of Iterates of Rational Maps on Normal Projective Varieties

Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh, Sibony [DS05b], and by Truong [Tru16].Precisely, we give a new proof of the submultiplicativity properties of these degrees and of its birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao [Xia15] and Popovici [Pop16], which generalizes Siu's inequality (see [Trap95]) to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in X.

math.AG↗
arXiv2019

Higher arithmetic degrees of dominant rational self-maps

Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $λ_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmetic degrees of $f$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

math.NT↗
arXiv2019

Positivity of valuations on convex bodies and invariant valuations by linear actions

In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to non-smooth valuations, which are in the completion of the spaces of valuations with respect to these norms. The novelty of our approach lies in the fact that our proof does not rely on the general theory of wave fronts, but on geometric inequalities deduced from optimal transport methods. We apply this result to prove a variant of Minkowski's existence theorem, and generalize a theorem of Favre-Wulcan and Lin in complex dynamics over toric varieties by studying the linear actions on the Banach spaces of valuations and by studying their corresponding eigenspaces.

math.DG↗
arXiv2018

Degree growth for tame automorphisms of an affine quadric threefold

In this paper, we consider the degree sequences of the tame automorphisms preserving an affine quadric threefold. Using some valuatives estimates derived from the work of Shestakov-Umirbaev and the action of this group on a CAT(0), Gromov-hyperbolic square complex constructed by Bisi-Furter-Lamy, we prove that the dynamical degrees of tame elements avoid any value strictly between 1 and 4/3. As an application, these methods allow us to characterize when the growth exponent of the degree of a random product of finitely many tame automorphisms is positive.

math.AG↗