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Register a new userOn quasi-invariant curves
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Authors
Ricardo Perez-Marco
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Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccozs complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a geometric interpretation of Denjoy-Yoccoz lemma using the hyperbolic metric that gives a direct construction of quasi-invariant curves without renormalization.
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In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon $eta$, we can construct an outer billiard table $T$ by cutting out a fixed area from the interior of $eta$. $T$ is piece-wise hyperbolic and the polygon $eta$ is an invariant curve of $T$ under the billiard map $phi$. We will show that, if $beta $ is a periodic point under the outer billiard map with rational rotation number $tau = p / q$, then the $n$th iteration of the billiard map is not the local identity at $beta$. This proves that the rotation number $tau$ as a function of the area parameter is a devils staircase function.
Let $f:Xto X$ be a dominating meromorphic map of a compact Kahler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $ u$ of large entropy supported by $mathrm{supp}(S)$. Defining dimensions for $ u$ and $S$, we give inequalities `a la Ma~ne involving the Lyapunov exponents of $ u$ and those dimensions. We give dynamical applications of those inequalities.
We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We determine all possible limits of RN differentials in degenerating sequences of smooth curves, and describe the limit in terms of solutions of the corresponding Kirchhoff problem. We further show that the limit of zeroes of RN differentials is the set of zeroes of a twisted meromorphic RN differential, which we explicitly construct. Our main new tool is an explicit solution of the jump problem on Riemann surfaces in plumbing coordinates, by using the Cauchy kernel on the normalization of the nodal curve. Since this kernel does not depend on plumbing coordinates, we are able to approximate the RN differential on a smooth plumbed curve by a collection of meromorphic differentials on the irreducible components of a stable curve, with an explicit bound on the precision of such approximation. This allows us to also study these approximating differentials at suitable scales, so that the limit under degeneration is not identically zero. These methods can be applied more generally to study degenerations of differentials on Riemann surfaces satisfying various conditions.
For a rational function f we consider the norm of the derivative with respect to the spherical metric and denote by K(f) the supremum of this norm. We give estimates of this quantity K(f) both for an individual function and for sequences of iterates.
Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian variety over $K$. Let us consider the canonical embedding of $C$ into $J$ that sends $O$ to the zero of the group law on $J$. This embedding allows us to identify $C(K)$ with a certain subset of the commutative group $J(K)$. A special case of the famous theorem of Raynaud (Manin--Mumford conjecture) asserts that the set of torsion points in $C(K)$ is finite. It is well known that the points of order 2 in $C(K)$ are exactly the remaining $(2g+1)$ Weierstrass points. One of the authors proved that there are no torsion points of order $n$ in $C(K)$ if $3le nle 2g$. So, it is natural to study torsion points of order $2g+1$ (notice that the number of such points in $C(K)$ is always even). Recently, the authors proved that there are infinitely many (for a given $g$) mutually nonisomorphic pairs $C,O)$ such that $C(K)$ contains at least four points of order $2g+1$. In the present paper we prove that (for a given $g$) there are at most finitely many (up to a isomorphism) pairs $(C,O)$ such that $C(K)$ contains at least six points of order $2g+1$.
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