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, $\mu_g=\mu_f$ 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.
Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton s...
Let $p$ be a prime. We provide examples which show that \'etale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\in\mathbb N\times\mathbb N^{\ast}$. We provide examples of such \'etale endomorphisms whose images have com...
It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-h...
Gao-Feng Huang, F. Kutzschebauch, Son Nam Tran et al.· 0 citations
Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary pro...
We prove the existence of $C^{r,1-}$-regular stable invariant manifolds and linear conjugacies for analytic maps near a hyperbolic fixed point in the presence of resonances. The regularity exponent $r$ depends on the minimal resonant index, while the H\"older exponent can be chosen arbitrarily close to $1$, i.e., $1-\v...
The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every...
Fei Hu, Jiang Chen· 0 citations
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