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We study canonical heights for plane polynomial mappings of small topologicaldegree. In particular, we prove that for points of canonical height zero, thearithmetic degree is bounded by the topological degree and hence strictlysmaller than the first dynamical degree. The proof uses the existence, provedby Favre and the first author, of certain compactifications of the planeadapted to the dynamics.

Canonical heights for plane polynomial maps of small topological degree