Daily Math Paper Digest
Arithmetic dynamics in the broad sense.
For every dominant self-map of an abelian variety, a Zariski-dense orbit has arithmetic degree equal to the map's dynamical degree. The proof separates the isogeny's generalized 1-eigenspace, where translated iterates have only polynomial height growth, from the complementary expanding part.
A uniform bound is proved for each rational preperiodic orbit of an endomorphism of the projective line over a global field, in terms of the field and the number of places of bad reduction but not the map's degree. The proof combines good-reduction distance identities, bounded residue fields, and quantitative two-variable S-unit equations to control both cycles and tails.
Ghioca and Mavraki prove that the canonical height of a rational section in the family $f_t(z)=(z^d+t)/z$ differs by a bounded amount from its generic canonical height times the parameter height. Their proof obtains a uniform place-by-place comparison even near the degree-dropping fibre at $t=0$.
Blanc, Canci, and Elkies determine the birational type of the moduli surfaces of quadratic rational maps with a marked cycle through period six. The period-six surface is of general type but nevertheless has infinitely many rational points, supplied by explicit rational and positive-rank elliptic curves.
A uniform height-growth bound for dominant rational self-maps shows that an orbit's upper arithmetic degree cannot exceed the map's dynamical degree. The same estimate also constructs canonical heights for divisor classes that are eigenvectors only up to algebraic equivalence.
A bounded-degree descent argument turns geometric conjugacy of twists into a uniform bound for rational preperiodic points. The paper also gives the sharp possibilities 2, 4, or 6 for the family z + b/z over Q.
Silverman proves that for monomial self-maps of an algebraic torus, points of canonical height zero lie in the divisible hull of a proper algebraic subgroup, and he determines all possible arithmetic degrees from the irreducible factors of the defining matrix's characteristic polynomial. The proof converts height vanishing into place-by-place linear relations and then uses Jordan theory, Baker's theorem, and Kronecker's theorem to descend them to integral characters.
Xie proves that a birational self-map of a projective surface with first dynamical degree greater than one has Zariski-dense non-critical periodic points over any algebraically closed field. The proof combines specialization of dynamical degree, Hrushovski's finite-field method, finiteness of periodic curves, and lifting through a discrete valuation ring.
Lin computes every dynamical degree of a dominant monomial map from the ordered moduli of the eigenvalues of its exponent matrix. The proof converts toric pullback into complementary minors and then reads exponential growth on an exterior power.
Amerik proves that a dominant rational self-map of infinite order over a number field has an algebraic point with infinite forward orbit, by combining good reduction, a periodic point over a finite field, and p-adic analytic interpolation.
Baker and DeMarco prove that two fixed complex points are simultaneously preperiodic for infinitely many maps z^d+c exactly when their d-th powers agree, using generalized Mandelbrot sets and adelic equidistribution over both number fields and function fields.
Manes proves that the moduli space of quadratic rational maps with a marked point of formal period N is geometrically irreducible for every N greater than one, while automorphisms force systematic reducibility on special dynamical loci.
Hutz constructs infinite families of quadratic polynomial morphisms of projective space with long rational periodic orbits, then combines relatively prime periods to obtain super-polynomial growth with the dimension.
Ih and Tucker prove that a non-preperiodic algebraic point has only finitely many Chebyshev-preperiodic points that are S-integral relative to it, using local equidistribution, Baker's theorem, and the global product formula.
For a rational map on the projective line over a number field, Petsche proves Ih's predicted finiteness of S-integral preperiodic points relative to a non-preperiodic point, under a place-by-place Fatou-set hypothesis.
Yap gives a quantitative archimedean form of Yuan's equidistribution theorem: outside a hypersurface of polynomially bounded degree, the discrepancy of a Galois orbit is controlled by canonical height and a chosen error scale. The proof converts regularized canonical metrics and arithmetic volume estimates into a small section whose divisor is the exceptional hypersurface.
For any dominant rational self-map over the algebraic numbers, points whose arithmetic degree comes arbitrarily close to the first dynamical degree are adelically dense. For birational maps with first dynamical degree larger than the third, this height-growth statement proves the Zariski dense orbit conjecture.
For fixed degrees, a Zariski-general pair of rational maps on the projective line has only uniformly many common preperiodic points. The proof combines a quantitative adelic energy inequality with nondegeneracy and equidistribution on the parameter space of pairs.
Cohomological hyperbolicity forces periodic points whose entire orbits remain in a suitable Zariski open set to have uniformly bounded height. The proof turns a big pullback divisor into a second-order height inequality whose expanding and contracting modes are incompatible with periodicity.
Outside a height-density-zero set of parameters, specialization preserves exactly the K-rational preperiodic portrait of a one-parameter rational map. The proof converts new portrait vertices to long cycles modulo thin sets, then traps long-cycle parameters in sparse residue classes using good reduction.
A quadratic rational map over Q with nonabelian automorphism group has no Q-rational cycle of period greater than 3 and at most six Q-rational preperiodic points. A root-of-unity argument excludes long cycles, while explicit dynatomic calculations control the remaining cycles and tails.
A polynomial's sparse homogeneous presentation yields a degree-uniform bound between its canonical and naive heights. The bound reduces rational preperiodic points for height-zero polynomials over Q to 0, +/-1, and +/-2, enabling a complete portrait classification.
Long Wang proves that periodic points of a cohomologically hyperbolic birational self-map have bounded height once their full orbits avoid a fixed proper exceptional subset. The proof converts a dynamical-degree gap into a two-sided height expansion using a big divisor on a resolution of indeterminacy.