Arithmetic intersection and measures of maximal entropy
Laura DeMarco - Harvard University
About 10 years ago, Xinyi Yuan and Shouwu Zhang proved that if two holomorphic maps f and g on P^N have the same sets of preperiodic points (or if the intersection of Preper(f) and Preper(g) is Zariski dense in P^N), then they must have the same measure of maximal entropy. This was new even in dimension N=1. I will describe some ingredients in their proof, while emphasizing the dynamical history behind this result. I will also sketch the proof of a theorem of Levin and Przytycki from the 1990s, in dimension N=1, that two (non-exceptional) maps have the same measure of maximal entropy if and only if they "essentially" share an iterate.
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