AI 中文总结
该文研究$\u2102^k$上正则多项式自同态的测度刚性，借助无穷远稳定流形结构与Grassmann流形上的可除性问题分析，证明了平衡测度相等的相关刚性结论及有限性结果，并给出二维情形的多类应用。
AI 中文摘要
我们证明了$\u2102^k$上正则多项式自同态的测度刚性结果。对于次数相同、其首齐次部分仅相差目标空间上一个可逆线性映射的映射，平衡测度相等等价于映射可通过Julia集的仿射对称进行后复合。随后我们证明，对于参数空间非空Zariski开子集中的映射$f$，无需对首项作相关假设，$\u03bc_g=\u03bc_f$即可推出$g=f$；在二维情形下，该结论对所有次数均成立，此时$g$是$f$的一个迭代。我们还在本质必要的假设下，建立了具有指定平衡测度的映射的有限性结果。\n我们的方法基于无穷远超平面附近的稳定流形结构。平衡测度相等会产生一族Zariski稠密的公共局部稳定流形芽，其内在一阶喷管导出了Grassmann流形上的可除性关联问题。我们通过分析幂映射纤维并应用一般稳定子论证解决了该问题。\n在二维情形下，应用包括基于预周期集的刻画、Tits型二择一定理、迭代中心化子相关结果，以及一个算术非稠密性定理。
英文摘要
We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show that, for $f$ in a nonempty Zariski open subset of the parameter space, $μ_g=μ_f$ implies $g=f$, with no hypothesis relating the leading terms; in dimension two this holds across all degrees, with $g$ an iterate of~$f$. We also establish finiteness results for maps with a prescribed equilibrium measure under essentially necessary hypotheses. Our approach rests on the stable manifold structure near the hyperplane at infinity. Equality of equilibrium measures yields a Zariski-dense family of common local stable-manifold germs, whose intrinsic first-order jets lead to a divisibility incidence problem on a Grassmannian. We resolve this problem by analyzing the power-map fiber and applying a generic stabilizer argument. In dimension two, applications include a characterization in terms of preperiodic sets, a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.
Comments47 pages, comments welcome
