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A Search for Additional Planets in the NASA EPOXI Observations of the Exoplanet System GJ 436

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]