Daily Math Paper Digest
Arithmetic dynamics in the broad sense.
The authors formulate small Arithmetic Non-Density (sAND), requiring bounded-degree points whose arithmetic degree is below the dynamical degree to lie in a proper closed locus, and prove it for several major geometric classes. A precise canonical-height description on abelian varieties explains how torsion translates and Northcott-type finiteness produce the non-density.
Matsuzawa proves Shibata's ample-canonical-height conjecture, hence the Kawaguchi--Silverman conjecture, for endomorphisms of normal projective varieties with finitely generated semi-ample nef cones and no numerical-versus-linear Picard discrepancy. The proof decomposes height growth along extremal nef rays and uses polarized canonical heights to trap the zero locus in a proper closed subset.
Lesieutre and Satriano construct a birational self-map of projective four-space with algebraic points realizing infinitely many distinct arithmetic degrees. The counterexample comes from special surface fibers whose dynamical degrees vary through an infinite sequence and whose birational models are automorphisms.
Sano constructs canonical heights for Jordan blocks with eigenvalues of modulus below one and uses them to prove a spectral-gap criterion: arithmetic degree one is equivalent to preperiodicity for a broad class of surjective endomorphisms.
Lin proves that every well-defined orbit of a dominant monomial self-map on a projective toric variety has an arithmetic degree, and that the possible values are exactly the spectral radii of the irreducible factors of the defining matrix's characteristic polynomial, together with 1. The proof turns toric boundary dynamics into periodic monomial dynamics on smaller tori.
For a surjective endomorphism of a smooth projective surface over an algebraic closure of a number field, every arithmetic degree exists, and a Zariski-dense orbit has arithmetic degree equal to the first dynamical degree. The proof combines birational and finite-cover reductions with the classification of surfaces admitting nontrivial endomorphisms.
For a dominant monomial self-map of projective space, Call and Silverman give an explicit dimension-dependent lower bound for its dynamical degree in terms of one of the first N degree-growth ratios. They also show how any such uniform estimate forces large drops of dynamical degree in a family to lie in a proper Zariski closed subset of the parameter space.
Ingram proves a Lang-type lower bound for canonical heights in weighted homogeneous polynomial families when the number of bad-reduction places is bounded. The proof turns local escape geometry into a global height inequality through a simultaneous pigeonhole argument over the bad places.
An irreducible complex plane curve contains infinitely many pairs of postcritically finite parameters for the unicritical family z^d+c only when it is a PCF coordinate line or a root-of-unity diagonal. The proof joins arithmetic equidistribution of PCF parameters to external-ray and wake rigidity for the generalized Mandelbrot set.
For a non-isotrivial algebraic family of rational maps on the projective line over a complex curve, dynamical stability of a marked point, canonical height zero, and preperiodicity are equivalent. The proof couples function-field height finiteness with a local escape-rate analysis at degenerate parameters.
Ingram proves a bounded-error specialization formula for canonical heights in fibral families of regular polynomial endomorphisms of projective space. The proof identifies the function-field escape rates with a divisor on the parameter curve by comparing local Green functions with local Weil heights.
Bell, Ghioca, and Tucker prove that orbit intersections with closed sets in Noetherian dynamics are a finite union of arithmetic progressions plus a Banach-density-zero remainder. This applies to every rational self-map whose chosen orbit avoids indeterminacy, over an arbitrary field.
Aluffi turns the multidegrees of generalized monomial rational maps into an integral over a Newton outer region, then into normalized simplex volumes. For a well-presented monomial self-map of projective space, the whole multidegree polynomial is the reversed characteristic polynomial of its torus exponent matrix.
Kawaguchi and Silverman construct canonical heights for divisor classes in a Jordan block and use them to prove that every orbit of a projective endomorphism has a well-defined arithmetic degree drawn from a finite set of algebraic integers. They also identify the zero set of a nef canonical height on an abelian variety and deduce the dense-orbit case of the arithmetic-degree conjecture there.
Lee constructs a semipositive adelic metric for strongly regular pairs of polynomial maps and applies Yuan's theorem to equidistribute small points. Degree-balanced iterates of a regular polynomial automorphism and its inverse then yield equidistribution of generic periodic points at every place.
Benedetto, Ghioca, Kurlberg, and Tucker prove a dynamical Mordell--Lang theorem for curves under the diagonal action of a rational map on a power of the projective line. Their proof combines arithmetic-surface intersection theory with p-adic analytic interpolation to force infinite hitting sets to contain full arithmetic progressions.
Abboud and Xie extend dynamical degrees to twisted rational maps, identify relative degrees with those of the generic-fibre dynamics, and prove a mixed-degree formula. They apply this framework to algebraicity bounds for the first dynamical degree of twisted affine endomorphisms.
De Thélin, Dinh, and Kaufmann prove exponential equidistribution of the repelling period-$n$ points of a holomorphic endomorphism of projective space that lie in the small Julia set and have quantitatively large multipliers. Their inverse-branch construction also shows that non-repelling points and points outside the small Julia set form an exponentially negligible fraction.
Young proves a uniform finiteness theorem for preperiodic points of monomial semigroups that are S-integral relative to a bounded-degree non-preperiodic point. The proof combines sequence-wise canonical heights and equidistribution with binomial factorization, linear forms in logarithms, and the product formula.
For a surjective endomorphism of a normal projective variety in characteristic zero, the arithmetic degree of any point with Zariski-dense orbit is one of the cohomological Lyapunov multipliers. The proof combines the Albanese factor, equivariant lifts of numerical eigenspaces, vector-valued canonical heights, and a big-divisor growth contradiction.
For any surjective endomorphism of a projective Q-factorial toric variety, every eigenvalue of the pullback on the Neron--Severi space whose modulus exceeds one is realized as the arithmetic degree of an algebraic point. The proof combines equivariant extremal contractions, preperiodic points on toric fibers, and canonical heights attached to nef eigendivisors.
For an endomorphism of projective N-space, Ingram gives completely explicit constants bounding the difference between the canonical and Philippon heights of every effective divisor, linearly in the divisor degree. Local escape-rate estimates for points and homogeneous forms, together with the product formula, turn this bound into an effective approximation algorithm for divisor canonical heights.
For normalized degree-d polynomials over the rationals, the average number of rational preperiodic points tends to zero with a power saving, while the smallest positive canonical height is generically asymptotic to log H divided by d(d-1). The proof turns local escape estimates at the superattracting point at infinity into counting restrictions on the k-free part of the constant denominator.
For polynomials with a finite superattracting cycle and nonarchimedean bad reduction, Looper proves a Bogomolov property over the maximal abelian extension. The paper also derives conditional number-field uniformity and function-field analogues, with non-isotriviality required for uniform boundedness, from quantitative equidistribution and filled-Julia-set geometry.
A unicritical complex polynomial whose every cycle multiplier is rational is affinely conjugate to a power map or a Chebyshev map. The proof encodes multipliers in resultant polynomials and reduces the exceptional quadratic case to an infinite descent in the Eisenstein integers.
For every surjective self-morphism of a projective variety over a number field with first dynamical degree greater than one, Sano and Shibata construct a Zariski-dense set of algebraic points of maximal arithmetic degree whose forward orbits are pairwise disjoint. Their proof combines a nef canonical height with a prime divisor in the field degree to separate each new orbit from all earlier ones.
DeMarco, Krieger, and Ye prove an effective uniform bound for the number of common preperiodic points of two distinct quadratic polynomials z^2+c. Their proof turns adelic energy estimates, quantitative equidistribution, and specialization into the explicit bound 10^103.
The Arakelov--Zhang pairing with the squaring map is expressed as a canonical height minus explicit local Julia-measure integrals. The formula yields a sharp Julia-set disjointness criterion, height bounds, explicit computations, and a rigidity theorem for integral polynomials.
Carney proves that two polarized dynamical systems over a one-variable function field have the same canonical-height-zero points on the Zariski closure of their common zero set. The proof constructs admissible adelic metrics and uses a function-field arithmetic Hodge index theorem whose equality case detects precisely the constant-field trace.
Lesieutre and Satriano prove the Kawaguchi--Silverman conjecture for every surjective endomorphism of a projective hyper-Kähler variety by constructing paired canonical heights from the expanding directions of an automorphism. The Beauville--Bogomolov form makes their sum big, while augmented-base-locus and Northcott arguments turn height growth into the predicted arithmetic degree.
Shibata normalizes the growth of an arbitrary ample height by both the dynamical degree and a polynomial factor, and proves a Northcott-type non-density statement for ample-eigendivisor maps, Picard-rank-two automorphisms, abelian varieties, and all smooth projective surfaces.
Doyle constructs a dynamical modular curve for every admissible preperiodic graph of a quadratic polynomial and proves that it is irreducible in characteristic zero. The same geometry yields a realization theorem over number fields and the full generic Galois group on preperiodic points.
For rational self-maps of the projective line over a number field with good reduction outside a finite set of places, the paper bounds rational periodic, tail, and preperiodic points by reducing nonarchimedean separation to S-unit and Thue--Mahler equations. It also proves degree-independent conditional bounds and shows that their numerical threshold hypotheses are sharp.
For every dominant rational self-map of a smooth projective variety over the algebraic numbers, the upper arithmetic degree of any well-defined orbit is at most the first dynamical degree. The proof resolves indeterminacy, controls numerically trivial height errors by square-root terms, and chooses a suitable iterate so the remaining polynomial loss is absorbed into an arbitrarily small exponential margin.
For a nonexceptional split rational dynamical system on the product of two projective lines, a transverse curve contains infinitely many coordinatewise preperiodic points precisely when suitable equal-degree iterates make the curve preperiodic. The proof turns equidistribution of small points into equality of invariant measures and then uses rigidity of local Julia-set symmetries.
A quantitative reduction theorem preserves the exact number of distinct geometric zeros of an integral polynomial system outside an explicitly bounded finite set of primes. Degree and height estimates for iterates turn this result into uniform bounds for periodic points and for the frequency with which finite-field orbits meet a variety.
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.