3,130,335 research outputs found

### On Semi-Periods

The periods of the three-form on a Calabi-Yau manifold are found as solutions
of the Picard-Fuchs equations; however, the toric varietal method leads to a
generalized hypergeometric system of equations which has more solutions than
just the periods. This same extended set of equations can be derived from
symmetry considerations. Semi-periods are solutions of this extended system.
They are obtained by integration of the three-form over chains; these chains
can be used to construct cycles which, when integrated over, give periods. In
simple examples we are able to obtain the complete set of solutions for the
extended system. We also conjecture that a certain modification of the method
will generate the full space of solutions in general.Comment: 18 pages, plain TeX. Revised derivation of $\Delta^*$ system of
equations; version to appear in Nuclear Physics

### Geometric Waldspurger periods II

In this paper we extend the calculation of the geometric Waldspurger periods
from our paper math/0510110 to the case of ramified coverings. We give some
applications to the study of Whittaker coefficients of the theta-lifting of
automorphic sheaves from PGL_2 to the metaplectic group Mp_2, they agree with
our conjectures from arXiv:1211.1596. In the process of the proof, we get some
new automorphic sheaves for GL_2 in the ramified setting. We also formulate
stronger conjectures about Waldspurger periods and geometric theta-lifting for
the dual pair (SL_2, Mp_2).Comment: 59 pages, final version, to appear in Representation theory
(electronic J. of AMS

### Periods implying almost all periods, trees with snowflakes, and zero entropy maps

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself,
$End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$.
We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a
cycle of period $n$, then $f$ has cycles of all periods greater than
$2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime
number greater than $End(X)$ and $f$ has cycles of all periods from 1 to
$2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture
of Misiurewicz for tree maps). Together with the spectral decomposition theorem
for graph maps it implies that $h(f)>0$ iff there exists $n$ such that $f$ has
a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree
maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the
period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd
integer with prime divisors less than $End(X)+1$

### Notes on Motivic Periods

The second part of a set of notes based on lectures given at the IHES in 2015
on Feynman amplitudes and motivic periods

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