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Let
$K$
be a finitely generated field of characteristic zero. For fixed
$m\geqslant 2$
, we study the rational functions
$\unicode[STIX]{x1D719}$
defined over
$K$
that have a
$K$
-orbit containing infinitely many distinct
$m$
-th powers. For
$m\geqslant 5$
we show that the only such functions are those of the form
$cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$
with
$\unicode[STIX]{x1D713}\in K(x)$
, and for
$m\leqslant 4$
we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before.
With additional analysis, we show that the index set
$\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$
is a union of finitely many arithmetic progressions, where
$\unicode[STIX]{x1D719}^{n}$
denotes the
$n$
-th iterate of
$\unicode[STIX]{x1D719}$
and
$\unicode[STIX]{x1D706}\in K(x)$
is any map Möbius-conjugate over
$K$
to
$x^{m}$
. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves
$y^{m}=\unicode[STIX]{x1D719}^{n}(x)$
. We describe all
$\unicode[STIX]{x1D719}$
for which these curves have an irreducible component of genus at most 1, and show that such
$\unicode[STIX]{x1D719}$
must have two distinct iterates that are equal in
$K(x)^{\ast }/K(x)^{\ast m}$
.
The work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.
For almost any compact connected Lie group
$G$
and any field
$\mathbb{F}_{p}$
, we compute the Batalin–Vilkovisky algebra
$H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$
on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if
$p$
is odd or
$p=0$
, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology
$HH^{\star }(H_{\star }(G),H_{\star }(G))$
. Over
$\mathbb{F}_{2}$
, such an isomorphism of Batalin–Vilkovisky algebras does not hold when
$G=\text{SO}(3)$
or
$G=G_{2}$
. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.
We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.
This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in (
$x,t$
)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.
Let
$(M^{n},g)$
be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in
$H^{\unicode[STIX]{x1D6FC}}(M)$
. For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that
$\unicode[STIX]{x1D6FC}>\frac{1}{2}$
is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound
$\unicode[STIX]{x1D6FC}>1$
from interpolation. We managed to go below 1 for dimension
${\leqslant}$
3. The more interesting thing is that, for a one-dimensional compact manifold,
$\unicode[STIX]{x1D6FC}>\frac{1}{3}$
is sufficient.