In 1993, Silverman proved that the coordinates of orbits on P^1 have asymptotically the same digits.

Behind this theorem, there is an estimate of growth of local height function on P^1 along the orbit.

I generalized the problem to higher dimensional dynamical systems on algebraic varieties involving local heights associated with closed subschemes.

I proved a generalization of Silverman's theorem, partially assuming Vojta's conjecture.

Let $X$ be a smooth projective variety defined over an algebraically closed field of arbitrary characteristic, and $f\colon X \to X$ a surjective morphism. The $i$-th cohomological dynamical degree $\chi_i(f)$ of $f$ is defined as the spectral radius of the pullback $f^*$ on the \'etale cohomology group $H^i_{et}(X, \bQ_\ell)$ and the $k$-th numerical dynamical degree $\lambda_k(f)$ as the spectral radius of the pullback $f^*$ on the vector space $N^k(X)_\bR$ of real algebraic cycles of codimension $k$ modulo numerical equivalence. Truong conjectured that $\chi_{2k}(f) = \lambda_k(f)$ for any $1 \le k \le \dim X$. When the ground field is the complex number field, the equality follows from the positivity property inside the de Rham cohomology of the ambient complex manifold $X(\bC)$. We prove this conjecture in the case of abelian varieties. The proof relies on a new result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

Given an endomorphism on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this talk, we see that there are densely many rational points with maximal arithmetic degree. This is a joint work with Kaoru Sano.

In this talk, I will show that, a rationally connected smooth projective n-fold is n-th product of rational curves if and only if it admits a surjective endomorphism f with its pullback action on the Néron-Severi space having n distinct eigenvalues greater than 1. This is a joint work with Dr. Sheng Meng.

In this talk, we will give a structure theorem of endomorphisms of normal projective surfaces via equivariant minimal model program. Combining with some local dynamics and known results, we will give an application of the main theorem to Zariski dense orbit property in dimension two. This is a joint work with Dr Junyi Xie and Professor De-Qi Zhang.

In this talk, we will discuss the structure of varieties admitting a special endomorphism called polarized endomorphism or int-amplified endomorphism. Fujimoto and Nakayama classified smooth surfaces admitting a non-invertible endomorphism using equivariant contractions of (-1)-curves. This strategy also works in the higher dimensional case if we assume the variety has a special endomorphism proved by Meng and Zhang. Improving this strategy, we obtain Fano type fibrations replacing the variety into an étale cover.