  We present time series photometry of the M dwarf transiting exoplanet system
GJ 436 obtained with the the EPOCh (Extrasolar Planet Observation and
Characterization) component of the NASA EPOXI mission. We conduct a search of
the high-precision time series for additional planets around GJ 436, which
could be revealed either directly through their photometric transits, or
indirectly through the variations these second planets induce on the transits
of the previously known planet. In the case of GJ 436, the presence of a second
planet is perhaps indicated by the residual orbital eccentricity of the known
hot Neptune companion. We find no candidate transits with significance higher
than our detection limit. From Monte Carlo tests of the time series, we rule
out transiting planets larger than 1.5 R_Earth interior to GJ 436b with 95%
confidence, and larger than 1.25 R_Earth with 80% confidence. Assuming
coplanarity of additional planets with the orbit of GJ 436b, we cannot expect
that putative planets with orbital periods longer than about 3.4 days will
transit. However, if such a planet were to transit, we rule out planets larger
than 2.0 R_Earth with orbital periods less than 8.5 days with 95% confidence.
We also place dynamical constraints on additional bodies in the GJ 436 system.
Our analysis should serve as a useful guide for similar analyses for which
radial velocity measurements are not available, such as those discovered by the
Kepler mission. These dynamical constraints on additional planets with periods
from 0.5 to 9 days rule out coplanar secular perturbers as small as 10 M_Earth
and non-coplanar secular perturbers as small as 1 M_Earth in orbits close in to
GJ 436b. We present refined estimates of the system parameters for GJ 436. We
also report a sinusoidal modulation in the GJ 436 light curve that we attribute
to star spots. [Abridged]
