The two verbs avocarse and abocarse are pronounced exactly alike, and for that reason Spanish speakers sometimes write one of them when they mean the other.
Abocarse is a verb used only in Latin America (not in Spain) and is a synonym of dedicarse. So se aboca a la redacción del contrato means "he is drafting the contract" (literally "he is dedicating himself to the drafting of the contract").
Avocarse, on the other hand, is used only in legal language and means "to exercise reach-down jurisdiction." An example would be La Corte Suprema se avocó el conocimiento de la causa, meaning "The Supreme Court exercised reach-down jurisdiction over the case" or "The case was removed to the Supreme Court."
Confusingly, in legal documents from Ecuador you often see a variation of this phrase: avocar el conocimiento (instead of avocarse el conocimiento) and in Ecuador, this phrase has nothing to do with "reach-down jurisdiction." Instead, in Ecuadorian Spanish, avocar el conocimiento de una causa is simply a longwinded way of saying conocer de una causa (to hear a case). So La Corte Nacional avocó el conocimiento de la causa is equivalent to saying La Corte Nacional conoció de la causa and means "The National Court heard the case." An essay explaining this unique Ecuadorian usage (which the author considers incorrect) can be found here.
Another example of unusual usage in Ecuadorian legal Spanish is the verb decurrir, which means "to run" (correr). So el mes que descurre means el mes en curso (the current month). Empezó a decurrir el plazo means empezó a correr el plazo (the period began to run). This verb even appears in the Ecuadorian Code of Civil Procedure. For example, Art. 324 states that No se aceptará la apelación ... antes de que empiece a decurrir el término fijado en el inciso anterior.... In other words, no appeal will be accepted until the period established in the previous paragraph begins to run.