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Tuyen Trung Truong

Publications and source records attributed to Tuyen Trung Truong.

72 records · Page 4Linked to original sources
arXiv2014

Salem numbers in dynamics of Kähler threefolds and complex tori

Let $X$ be a compact Kähler manifold of dimension $k\leq 4$ and $f:X\rightarrow X$ a pseudo-automorphism. If the first dynamical degree $λ_1(f)$ is a Salem number, we show that either $λ_1(f)=λ_{k-1}(f)$ or $λ_1(f)^2=λ_{k-2}(f)$. In particular, if $\mbox{dim}(X)=3$ then $λ_1(f)=λ_2(f)$. We use this to show that if $X$ is a complex 3-torus and $f$ is an automorphism of $X$ with $λ_1(f)>1$, then $f$ has a non-trivial equivariant holomorphic fibration if and only if $λ_1(f)$ is a Salem number. If $X$ is a complex 3-torus having an automorphism $f$ with $λ_1(f)=λ_2(f)>1$ but is not a Salem number, then the Picard number of $X$ must be 0,3 or 9, and all these cases can be realized.

math.AG↗
arXiv2013

Some dynamical properties of pseudo-automorphisms in dimension 3

Let $X$ be a compact Kähler manifold of dimension 3 and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $λ_1(f)^2>λ_2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be proved for pseudo-automorphisms in dimension 3. As a byproduct, we show that the intersection of some dynamically related currents are well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.

math.DS↗
arXiv2013

Unirationality of Ueno-Campana's threefold

We shall prove that the threefold studied in the paper "Remarks on an Example of K. Ueno" by F. Campana is unirational. This gives an affirmative answer to a question posed in the paper above and also in the book by K. Ueno, "Classification theory of algebraic varieties and compact complex spaces".

math.AG↗
arXiv2013

Geometric properties of upper level sets of Lelong numbers on projective spaces

Let $T$ be a positive closed current of unit mass on the complex projective space $\mathbb P^n$. For certain values $α<1$, we prove geometric properties of the set of points in $\mathbb P^n$ where the Lelong number of $T$ exceeds $α$. We also consider the case of positive closed currents of bidimension (1,1) on multiprojective spaces.

math.CV↗
arXiv2013

Equidistribution for meromorphic maps with dominant topological degree

Let f be a meromorphic self-map on a compact Kaehler manifold whose topological degree is strictly larger than the other dynamical degrees. We show that repelling periodic points are equidistributed with respect to the equilibrium measure of f. We also describe the exceptional set of points whose backward orbits are not equidistributed.

math.DS↗
arXiv2013

On automorphisms of blowups of projective manifolds

In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\geq 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\rightarrow X_0$ is a finite blowup along smooth centers, here $X_0$ is a projective manifold of interest. Here we allow $X_0$ to be either one of the following manifolds: it has Picard number 1, or a Fano manifold, or it is a projective hyper-Kähler manifold. We also allow the centers of blowups to have large dimensions relative to that of $X_0$ (may be upto $dim(X_0)-2$). Explicit constructions are given in Section \ref{SectionBlowupsAndNonVanishingConditions}, where we also show that the assumptions in the results in that section are necessary (see Example 6 in Section \ref{SectionBlowupsAndNonVanishingConditions}). As a consequence, we obtain new examples of manifolds $X$, whose any automorphism is either of zero topological entropy or is cohomologically hyperbolic.

math.DS↗
arXiv2012

On automorphisms of blowups of $\mathbb{P}^3$

Let $π:X\rightarrow \mathbb{P}^3$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow \mathbb{P}^3$, whose automorphism group $Aut(X)$ has only finitely many connected components. We also present a heuristic argument showing that for a "generic" compact Kähler manifold $X$ of dimension $\geq 3$, the automorphism group $Aut(X)$ has only finitely many connected components.

math.DS↗
arXiv2012

The simplicity of the first spectral radius of a meromorphic map

Let $X$ be a compact Kähler manifold and let $f:X\rightarrow X$ be a dominant rational map which is 1-stable. Let $λ_1$ and $λ_2$ be the first and second dynamical degrees of $f$. If $λ_1^2>λ_2$, then we show that $λ_1$ is a simple eigenvalue of $f^*:H^{1,1}(X)\rightarrow H^{1,1}(X)$, and moreover the unique eigenvalue of modulus $>\sqrt{λ_2}$. A variant of the result, where we consider the first spectral radius in the case the map $f$ may not be 1-stable, is also given. An application is stated for bimeromorphic selfmaps of 3-folds. In the last section of the paper, we prove analogs of the above results in the algebraic setting, where $X$ is a projective manifold over an algebraic closed field of characteristic zero, and $f:X\rightarrow X$ is a rational map. Part of the section is devoted to defining dynamical degrees in the algebraic setting. We stress that here the dynamical degrees of rational maps can be defined over any algebraic closed field, not necessarily of characteristic zero.

math.DS↗
arXiv2011

Pull-back of currents by meromorphic maps

Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\rightarrow Y$ be a dominant meromorphic map. Base upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{\sharp}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus for {\it any} current, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can be readily extended to the case of meromorphic correspondences. We give an example of a meromorphic map $f$ and two nonzero positive closed currents $T_1,T_2$ for which $f^{\sharp}(T_1)=-T_2$. We use Siu's decomposition to help further study on pulling back positive closed currents. Many applications on finding invariant currents are given.

math.DS↗
arXiv2011

Quasi-potentials and regularization of currents, and applications

Let $Y$ be a compact Kähler manifold. We show that the weak regularization $K_n$ of Dinh and Sibony for the diagonal $Δ_Y$ (see Section 2 for more detail) is compatible with wedge product in the following sense: If $T$ is a positive $dd^c$-closed $(p,p)$ current and $θ$ is a smooth $(q,q)$ form then there is a sequence of positive $dd^c$-closed $(p+q,p+q)$ currents $S_n$ whose masses converge to 0 so that $-S_n\leq K_n(T\wedge θ)-K_n(T)\wedge θ\leq S_n$ for all $n$. We also prove a result concerning the quasi-potentials of positive closed currents. We give two applications of these results. First, we prove a corresponding compatibility with wedge product for the pullback operator defined in our previous paper. Second, we define an intersection product for positive $dd^c$-closed currents. This intersection is symmetric and has a local nature.

math.DS↗
arXiv2011

On the dynamical degrees of meromorphic maps preserving a fibration

Let f be a dominant meromorphic self-map on a compact Kaehler manifold X which preserves a fibration given by a meromorphic map from X to a compact Kaehler manifold Y. We compute the dynamical degrees of f in term of its dynamical degrees relative to the fibration and the dynamical degrees of the map g on Y which is induced by f. We derive from this result new properties of some fibrations intrinsically associated to X when this manifold admits an interesting dynamical system.

math.DS↗
arXiv2010

Degree complexity of birational maps related to matrix inversion: Symmetric case

For $q\geq 3$, we let $\mathcal{S}_q$ denote the projectivization of the set of symmetric $q\times q$ matrices with coefficients in $\mathbb{C}$. We let $I(x)=(x_{i,j})^{-1}$ denote the matrix inversion, and we let $J(x)=(x_{i,j}^{-1})$ be the matrix whose entries are the reciprocals of the entries of $x$. We let $K|\mathcal{S}_q=I\circ J:\mathcal{S}_q\rightarrow \mathcal{S}_q$ denote the restriction of the composition $I\circ J$ to $\mathcal{S}_q$. This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity of $K|\mathcal{S}_q$, thus confirming a conjecture of Angles d'Auriac, Maillard, and Viallet in [J. Phys. A: Math. Gen. 39 (2006), 3641--3654].

math.DS↗
arXiv2009

Local growth of pluri-subharmonic functions

We obtain two-bound estimates for the local growth of pluri-subharmonic functions in terms of Siciak and relative extremal functions. As applications, we give simple new proofs of "Bernstein doubling inequality" and the main result in [Alexander Brudnyi, Local inequalities for pluri-subharmonic functions, Annals Math. 149 (1999), No. 2, pp. 511--533]. We propose a conjecture similar to the comparison theorem in [H. Alexander and B. A. Taylor, Comparison of two capacities in $\mathbb{C}^n$, Math. Z. 186 (1984), 407--417], whose validity allows to obtain bounds for the local growth of pluri-subharmonic functions solely in term of the Siciak extremal functions.

math.CV↗
arXiv2009

Degree Complexity of Matrix Inversion

For a q by q matrix x=(x_{i,j}) we let J(x)=(x_{i,j}^{-1}) be the Hadamard inverse, which takes the reciprocal of the elements of x . We let I(x)=(x_{i,j})^{-1} denote the matrix inverse, and we define K=I\circ J to be the birational map obtained from the composition of these two involutions. We consider the iterates K^n=K\circ...\circ K and determine degree complexity of K, which is the exponential rate of degree growth of the degrees of the iterates.

math.DS↗
arXiv2008

Note on potential theory for functions in Hardy classes

The purpose of this note is to show that the set functions defined in \cite{trong-tuyen} can be suitably extended to all subsets $E$ of the unit disk $\mathbb{D}$. In particular we obtain uniform nearly-optimal estimates for the following quantity D_p(E,ε, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, ||g||_{H^p}\leq 1, (1-|ζ|)|g(ζ)| \leq ε\forall ζ\in E\}.

math.CV↗
arXiv2008

Nearly-optimal estimates for the stability problem in Hardy spaces

We continue the work of \cite{TLNT}. Let $E$ be a non-Blaschke subset of the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. Fixed $1\leq p\leq \infty$, let $H^p(\mathbb{D})$ be the Hardy space of holomorphic functions in the disk whose boundary value function is in $L^p(\partial \mathbb{D})$. Fixed $0 0$ define C_p(\varepsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(ζ)| \leq \varepsilon \forall ζ\in E\}. In this paper we find upper and lower bounds for $C_p(ε, R)$ when $ε$ is small for any non-Blaschke set $E$. The bounds are nearly-optimal for many such sets $E$, including sets contained in a compact subset of $\mathbb{D}$ and sets contained in a finite union of Stolz angles.

math.CV↗