Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.06211

Equilibrium webs and stability for arbitrary families of endomorphisms of projective spaces

Fabrizio Bianchi·Zhuchao Ji·Junyi Xie
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider arbitrary holomorphic families of endomorphisms of a complex projective space. We prove that the existence of an equilibrium web implies the holomorphic motion of all repelling cycles contained in the Julia set, i.e., the support of the unique measure of maximal entropy. Together with previous results of Berteloot, Bianchi, Brévard, Buff, and Dupont, this completes the extension of the stability-bifurcation theory to arbitrary families of holomorphic endomorphisms. The same conclusions also hold for holomorphic families of polynomial-like maps of large topological degree.

Explore related subjects

math.DSmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fabrizio Bianchi, Zhuchao Ji, Junyi Xie. 2026-11-04. Equilibrium webs and stability for arbitrary families of endomorphisms of projective spaces. https://arxiv.org/abs/2611.06211

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

KEEP EXPLORING

Related papers

arXiv2026

Metastability Landscapes in Nonlinear Ghost-Normal Scalar Fields

We investigate finite time dynamics in nonlinear classical scalar field systems containing one ghost and one normal kinetic sector. Using a spacetime finite element formulation, we study the dependence of threshold crossing and numerical continuation times on initial amplitude, characteristic wavenumber, spectral tilt, phase structure, and spatial localization. In the quartic model, larger amplitudes generally lead to earlier growth, while higher wavenumbers and ultraviolet-biased spectra favor longer resolved bounded evolution. Smooth compact initial data exhibit a broad region with no observed growth threshold crossing over the simulated interval. A direct comparison of two Gaussian evolutions shows that enhanced low-wavenumber power can precede rapid field growth, although this observation does not establish a universal spectral-transfer mechanism. We also compare the quartic interaction with a lifted $ϕ^6$ potential. The latter gives a markedly different continuation landscape, with a broad long-converged region followed by shorter accessible terminal times at larger amplitudes. Localization diagnostics do not identify a persistent oscillon core in the selected cases. By distinguishing amplitude growth, spectral reorganization, solver-limited termination, and finite-time censoring, our results characterize a nontrivial metastability landscape without implying global or quantum stability.

math.DS↗
arXiv2026

Slow Entropy of Quasi-Unipotent Actions on $S$-Arithmetic Homogeneous Spaces

We study metric and topological slow entropy for translations on a class of finite-volume $S$-arithmetic homogeneous spaces with real and non-Archimedean factors. We consider translations whose real component is quasi-unipotent and whose non-Archimedean component lies in a compact open subgroup. At polynomial scales, we show that the metric and topological slow entropies coincide and are determined by the chain structure of the real adjoint action. In particular, the non-Archimedean component contributes no additional polynomial slow entropy. Our result extends the slow entropy formula of Kanigowski, Vinhage, and Wei for quasi-unipotent flows on real homogeneous spaces to this $S$-arithmetic setting.

math.DS↗
arXiv2026

Discrete history operators: Exact state-space realization for sequence memory

A prescribed finite-dimensional analysis of an evolving discrete history need not admit an exact recurrence driven only by its current state and the entering input. We study exact realizations of causal linear discrete history operators (DHOs). Operator factorization and kernel inclusion characterize exact closure, while the cross-projection defect characterizes closure failure for orthonormal analyses. For finite-dimensional history spaces over fixed finite horizons, we distinguish two minimality problems. Reachable task quotients determine the minimum state dimension for preserving prescribed future tasks on reachable histories. Backward invariant envelopes characterize the minimum dimension of exact linear realizations recovering prescribed analyses on full history spaces within the admissible class. The minimum number of additional independent state directions is given by the rank of the propagated closure defect operator restricted to the kernel of the current analysis. For geometrically weighted countable histories, matrix mode transport yields structured exact recurrences and a uniform exponential decay estimate for state transitions. Additional conditions provide uniform Gram bounds and refined decay estimates. A stationary Meixner specialization yields explicit recurrence matrices, an O(N) arithmetic state update, and minimum augmentation dimensions over finite horizons for selected coefficients. Numerical experiments examine recurrence accuracy, state transition bounds, Gram conditions, and minimum realization dimensions.

math.DS↗