Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.09275

Dynamical degrees of twisted rational maps

Marc Abboud·Junyi Xie
View original source ↗Read source document ↗Export metadata ↓

Abstract

Twisted rational maps arise naturally in relative algebraic dynamics: if a rational self-map preserves a fibration, then the induced map on the generic fiber is usually not an ordinary rational self-map over the function field, but a twisted one. This suggests that twisted rational maps form a natural framework for studying relative dynamics. In this framework we extend the theory of dynamical degrees, the numerical invariants measuring the asymptotic complexity of a dynamical system: the defining limits exist, are independent of the choice of polarization, and are birational invariants. We also identify the relative dynamical degrees of a semi-conjugacy with the dynamical degrees of the induced twisted rational map on the generic fiber, and prove the corresponding mixed degree formula. Finally, using the spectral interpretation of dynamical degrees and valuative methods at infinity, we prove an algebraicity result for the first dynamical degree of twisted endomorphisms of affine varieties.

Explore related subjects

math.AGmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marc Abboud, Junyi Xie. 2026-08-10. Dynamical degrees of twisted rational maps. https://arxiv.org/abs/2608.09275

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lawson--Deligne Classes and Applications

We construct the integral Lawson--Deligne map of weight $q=n-p-k-1$ on smooth complex projective $n$-folds using filtered currents. It lifts the Friedlander--Mazur cycle class, recovers the reduced generalized Abel--Jacobi invariant on homologically trivial classes, and is compatible with algebraic correspondences. A Picard--Fuchs separation argument applied to the conic and van Geemen normal functions on the mirror quintic determines explicit regulator subspaces modulo the full rational period group. For prescribed elliptic moduli and a suitable mirror-quintic fiber, the subspace generated by their $a$- and $b$-loop products has dimension twice the $\Q$-dimension of the period-monomial space. Moduli $i\sqrt{\ell_j}$ for distinct primes $\ell_j$ give $2^{k+1}$ independent images on varieties of dimension $p+k+2$; one repeated imaginary quadratic modulus gives dimension four for every $k\geq1$. Compatibility with known projective-bundle and blow-up decompositions yields independent exceptional subspaces on smooth rational varieties. We also compare the higher Chow composite with the Bloch--KLM regulator after lowering the Hodge filtration. The KLM representative reduces to a cut-current class, and equality with the Lawson composite is proved in degree zero and for constant-unit decomposable classes. The general positive-degree comparison is reduced to an explicit filtered-realization condition.

math.AG↗
arXiv2026

Border rank lower bounds for families of GL(V)-invariant tensors

We give non-trivial lower bounds for the border rank of families of $\mathbf{GL}(V)$-invariant tensors in $U\otimes \mathbf{S}_λV\otimes \mathbf{S}_μV$ where $U$ is $V$, $\mathrm{Sym}^2V$ or $\bigwedge^2V$. In particular, we provide a family of tensors with border rank reaching arbitrarily close to $2\ell$ in the unbalanced case, where $\ell$ is the largest ambient vector space dimension. We do this by resolving a conjecture introduced by Wu, and obtaining new results on $6j$-symbols as a byproduct. We then generalize our results to $\mathrm{Sym}^2V$ and $\bigwedge^2 V$ using novel techniques based on an application of a theorem of Kostant and Kempf collapsing.

math.AG↗
arXiv2026

Maximally nodal sextic surfaces and linear determinantal representations

We prove that every maximally nodal sextic surface (with 65 nodes) $X \subset \mathbb{P}_{\mathbb{C}}^3$ contains a symmetric half-even set of nodes of cardinality 35. It follows that the associated half-quadratic sheaf is the cokernel of a symmetric $6 \times 6$ matrix of linear forms, yielding a linear determinantal representation of $X$. In particular, after a suitable Serre twist, the half-quadratic sheaf is an Ulrich sheaf of rank 1. As an example, we exhibit an explicit $6 \times 6$ matrix of linear forms whose determinant defines the Barth sextic surface.

math.AG↗