Post-starburst galaxies can be identified via the presence of prominent Hydrogen Balmer absorption lines in their spectra. We present a comprehensive study of the origin of strong Balmer lines in a volume-limited sample of 189 galaxies with $0.01<z<0.05$, $log(mbox{M}_{star}/mbox{M}_{odot})>9.5$ and projected axis ratio $b/a>0.32$. We explore their structural properties, environments, emission lines and star formation histories, and compare them to control samples of star-forming and quiescent galaxies, and simulated galaxy mergers. Excluding contaminants, in which the strong Balmer lines are most likely caused by dust-star geometry, we find evidence for three different pathways through the post-starburst phase, with most events occurring in intermediate-density environments: (1) a significant disruptive event, such as a gas-rich major merger, causing a starburst and growth of a spheroidal component, followed by quenching of the star formation (70% of post-starburst galaxies at $9.5<log(mbox{M}_{star}/mbox{M}_{odot})<10.5$ and 60% at $log(mbox{M}_{star}/mbox{M}_{odot})>10.5$); (2) at $9.5<log(mbox{M}_{star}/mbox{M}_{odot})<10.5$, stochastic star formation in blue-sequence galaxies, causing a weak burst and subsequent return to the blue sequence (30%); (3) at $log(mbox{M}_{star}/mbox{M}_{odot})>10.5$, cyclic evolution of quiescent galaxies which gradually move towards the high-mass end of the red sequence through weak starbursts, possibly as a result of a merger with a smaller gas-rich companion (40%). Our analysis suggests that AGN are `on for $50%$ of the duration of the post-starburst phase, meaning that traditional samples of post-starburst galaxies with strict emission line cuts will be at least $50%$ incomplete due to the exclusion of narrow-line AGN.
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