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In this paper we describe a uniform analysis of eight transits and elevensecondary eclipses of the extrasolar planet GJ 436b obtained in the 3.6, 4.5,and 8.0 micron bands using the IRAC instrument on the Spitzer Space Telescopebetween UT 2007 June 29 and UT 2009 Feb 4. We find that the best-fit transitdepths for visits in the same bandpass can vary by as much as 8% of the total(4.7 sigma significance) from one epoch to the next. Although we cannotentirely rule out residual detector effects or a time-varying, high-altitudecloud layer in the planet's atmosphere as the cause of these variations, weconsider the occultation of active regions on the star in a subset of thetransit observations to be the most likely explanation. We reconcile thepresence of magnetically active regions with the lack of significant visible orinfrared flux variations from the star by proposing that the star's spin axisis tilted with respect to our line of sight, and that the planet's orbit istherefore likely to be misaligned. These observations serve to illustrate thechallenges associated with transmission spectroscopy of planets orbitinglate-type stars; we expect that other systems, such as GJ 1214, may displaycomparably variable transit depths. Our measured 8 micron secondary eclipsedepths are consistent with a constant value, and we place a 1 sigma upper limitof 17% on changes in the planet's dayside flux in this band. Averaging over theeleven visits gives us an improved estimate of 0.0452% +/- 0.0027% for thesecondary eclipse depth. We combine timing information from our observationswith previously published data to produce a refined orbital ephemeris, anddetermine that the best-fit transit and eclipse times are consistent with aconstant orbital period. [ABRIDGED]

A Spitzer Transmission Spectrum for the Exoplanet GJ 436b, Evidence for Stellar Variability, and Constraints on Dayside Flux Variations