The source responsible for reionizing the universe at z > 6 remains uncertain. While an energetically adequate population of star-forming galaxies may be in place, it is unknown whether a large enough fraction of their ionizing radiation can escape into the intergalactic medium. Attempts to measure this escape-fraction in intensely star-forming galaxies at lower redshifts have largely yielded upper limits. In this paper we present new HST COS and archival FUSE far-UV spectroscopy of a sample of eleven Lyman Break Analogs (LBAs), a rare population of local galaxies that strongly resemble the high-z Lyman Break galaxies. We combine these data with SDSS optical spectra and Spitzer photometry. We also analyze archival FUSE observations of fifteen typical UV-bright local starbursts. We find evidence of small covering factors for optically-thick neutral gas in 3 cases. This is based on two independent pieces of evidence: a significant residual intensity in the cores of the strongest interstellar absorption-lines tracing neutral gas and a small ratio of extinction-corrected H-alpha to UV plus far-IR luminosities. These objects represent three of the four LBAs that contain a young, very compact (~100pc), and highly massive (~10^9 Mo) dominant central object (DCO). These three objects also differ from the other galaxies in showing a significant amount of blueshifted Ly-alpha emission, which may be related to the low covering factor of neutral gas. All four LBAs with DCOs in our sample show extremely high velocity outflows of interstellar gas, with line centroids blueshifted by about 700km/s and maximum outflow velocities reaching at least 1500km/s. We show that these properties are consistent with an outflow driven by a powerful starburst that is exceptionally compact. We speculate that such extreme feedback may be required to enable the escape of ionizing radiation from star forming galaxies.
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