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Junyi Xie

Publications and source records attributed to Junyi Xie.

At least 37 records · Page 2Linked to original sources

HiCoTraj:Zero-Shot Demographic Reasoning via Hierarchical Chain-of-Thought Prompting from Trajectory

Inferring demographic attributes such as age, sex, or income level from human mobility patterns enables critical applications such as targeted public health interventions, equitable urban planning, and personalized transportation services. Existing mobility-based demographic inference studies heavily rely on large-scale trajectory data with demographic labels, leading to limited interpretability and poor generalizability across different datasets and user groups. We propose HiCoTraj (Zero-Shot Demographic Reasoning via Hierarchical Chain-of-Thought Prompting from Trajectory), a framework that leverages LLMs' zero-shot learning and semantic understanding capabilities to perform demographic inference without labeled training data. HiCoTraj transforms trajectories into semantically rich, natural language representations by creating detailed activity chronicles and multi-scale visiting summaries. Then HiCoTraj uses a novel hierarchical chain of thought reasoning to systematically guide LLMs through three cognitive stages: factual feature extraction, behavioral pattern analysis, and demographic inference with structured output. This approach addresses the scarcity challenge of labeled demographic data while providing transparent reasoning chains. Experimental evaluation on real-world trajectory data demonstrates that HiCoTraj achieves competitive performance across multiple demographic attributes in zero-shot scenarios.

cs.AI↗

Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

The aims of this paper are to answer several conjectures and questions about multiplier spectrum of rational maps and to give new proofs of several rigidity theorems in complex dynamics, by combining tools from complex and non-archimedean dynamics. A remarkable theorem due to McMullen asserts that aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston's rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen's theorem without using quasiconformal maps or Teichmüller theory. We show that aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen's theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik. We show that a rational map is exceptional if and only if one of the following holds (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field; (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.

math.DS↗

Local rigidity of Julia sets

We find criteria ensuring that a local (holomorphic, real analytic, $C^1$) homeomorphism between the Julia sets of two given rational functions comes from an algebraic correspondence. For example, we show that if there is a local $C^1$-symmetry between the maximal entropy measures of two rational functions, then probably up to a complex conjugation, the two rational functions are dynamically related by an algebraic correspondence. The holomorphic case of our criterion will play an important role in the authors' upcoming proof of the Dynamical André-Oort conjecture for curves.

math.DS↗

The multiplier spectrum morphism is generically injective

In this paper, we consider the multiplier spectrum of periodic points, which is a natural morphism defined on the moduli space of rational maps on the projective line. A celebrated theorem of McMullen asserts that aside from the well-understood flexible Lattès family, the multiplier spectrum morphism is quasi-finite. In this paper, we strengthen McMullen's theorem by showing that the multiplier spectrum morphism is generically injective. This answers a question of McMullen and Poonen.

math.DS↗

Arithmetic degree and its application to Zariski dense orbit conjecture

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical degree of $f$. As an application, we prove that Zariski dense orbit conjecture holds for a birational map defined over $\overline{\mathbb{Q}}$ whose first dynamical degree is strictly larger than its third dynamical degree. In particular, the conjecture holds for birational maps on threefolds whose first dynamical is degree greater than $1$.

math.AG↗

Arithmetic Degrees are Cohomological Lyapunov Multipliers

For endomorphisms of projective varieties, we prove that the arithmetic degree of a point with Zariski dense orbit must be a cohomological Lyapunov multiplier of the dynamical system. We will apply our result to deduce a corollary towards the dynamical Mordell--Lang conjecture.

math.DS↗

Numerical spectrums control Cohomological spectrums

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $τ: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every $i=0,\dots,2d$, the spectral radius of $f^*$ on the numerical group $N^i(X)\otimes \mathbb{R}$ and on the $l$-adic cohomology group $H^{2i}(X_{\overline{\mathbf{k}}},\mathbb{Q}_l)\otimes \mathbb{C}$ are the same. As a consequence, if $f$ is $q$-polarized for some $q>1$, we show that the norm of every eigenvalue of $f^*$ on the $j$-th cohomology group is $q^{j/2}$ for all $j=0,\dots, 2d.$ This generalizes Deligne's theorem for Weil's Riemann Hypothesis to arbitary polarized endomorphisms and proves a conjecture of Tate. We also get some applications for the counting of fixed points and its ``moving target" variant. Indeed we studied the more general actions of certain cohomological coorespondences and we get the above results as consequences in the endomorphism setting.

math.AG↗

Algebraic dynamics and recursive inequalities

We get three basic results in algebraic dynamics: (1). We give the first algorithm to compute the dynamical degrees to arbitrary precision. (2). We prove that for a family of dominant rational self-maps, the dynamical degrees are lower semi-continuous with respect to the Zariski topology. This implies a conjecture of Call and Silverman. (3). We prove that the set of periodic points of a cohomologically hyperbolic rational self-map is Zariski dense. Moreover, we show that, after a large iterate, every degree sequence grows almost at a uniform rate. This property is not satisfied for general submultiplicative sequences. Finally, we prove the Kawaguchi-Silverman conjecture for a class of self-maps of projective surfaces including all the birational ones. In fact, for every dominant rational self-map, we find a family of recursive inequalities of some dynamically meaningful cycles. Our proofs are based on these inequalities.

math.DS↗

Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture

Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$. Assuming the standard conjecture $D$, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of semisimplicity of Frobenius endomorphism and of any polarized endomorphism (a more general result, in terms of the biggest size of Jordan blocks, holds). We illustrate these results through examples, including varieties dominated by rational maps from Abelian varieties and suitable products of $K3$ surfaces. Using the same idea, we provide a new proof of the main result in a recent paper by the third author, including Tate's question/Serre's conjecture that for a polarized endomorphism $f:X\rightarrow X$, all eigenvalues of the action of $f$ on $H^k(X)$ have the same absolute value.

math.AG↗

Numerical action for endomorphisms

Let $f: X\to X$ be a surjective endomorphism of a projective variety of dimension $d$. The aim of this paper is to study the action of $f$ on the numerical group of divisors. For exmaple, I proved that $f$ is cohomologically hyperbolic if and only if it is quasi-amplified; and it is amplified if and only if every subsystem of $(X,f)$ is cohomologically hyperbolic. For the proofs, I introduced a notion of spectrum in linear algebra for an open and saliant invariant cone. I also introduce a notion of generated (positive) cycles as an algebraic analogy of (positive) closed current.

math.DS↗

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures

Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $α_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $\barκ(X)$ is non-negative (resp. the étale fundamental group $π_1^{\text{ét}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with ''larger'' first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).

math.AG↗

The moduli space of a rational map is Carathéodory hyperbolic

Let $f$ be a rational map of degree $d\geq 2$. The moduli space $\mathcal{M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:\mathcal{M}_f\to X_f$ such that $i(\mathcal{M}_f)$ is precompact in $X_f$. In particular $\mathcal{M}_f$ is Carathéodory hyperbolic (i.e. bounded holomorphic functions separate points in $\mathcal{M}_f$), provided that $f$ is not flexible Lattès. This solves a conjecture of McMullen. When $d\geq 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.

math.CV↗

DAO for curves

We prove the Dynamical André-Oort (DAO) conjecture proposed by Baker and DeMarco for families of rational maps parameterized by an algebraic curve. In fact, we prove a stronger result, which is a Bogomolov type generalization of DAO for curves.

math.DS↗

The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties

In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of constructing entire curves in the pre-sequel "Partial heights, entire curves, and the geometric Bombieri-Lang conjecture." A new ingredient is an explicit description of the entire curves in terms of Lie algebras of abelian varieties.

math.NT↗

Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture

We introduce a new approach to the geometric Bombieri--Lang conjecture for hyperbolic varieties in characteristic 0. The main idea is to construct an entire curve on a special fiber of a variety over a complex function field from an infinite sequence of rational points of the variety. The construction relies on the classical Brody lemma in complex geometry and is conditional on a non-degeneracy conjecture of partial heights. In particular, we prove the geometric Bombieri-Lang conjecture for hyperbolic projective varieties which have finite morphisms to abelian varieties over function fields of characteristic 0.

math.NT↗

Around the dynamical Mordell-Lang conjecture

There are three aims of this note. The first one is to report some advances around the dynamical Mordell-Lang (=DML) conjecture. Second, we generalize some known results. For example, the Dynamical Mordell-lang conjecture was known for endomorphisms of $\mathbb{A}^2$ over $\overline{\mathbb{Q}}$. We generalize this result to all endomorphisms of $\mathbb{A}^2$ over $\mathbb{C}$. We generalize the weak DML theorem to a uniform version and to a version for partial orbit. Using this, we give a new proof of the Kawaguchi-Silverman-Matsuzawa' upper bound for arithmetic degree. We indeed prove a uniform version which works in both number field and function field case in any characteristic and it works for partial orbits. We also reformulate the ``$p$-adic method", in particular the $p$-adic interpolation lemma in language of Berkovich space and get more general statements. The third aim is to propose some further questions.

math.NT↗