A bit of clarification: there was no Maya "empire." Rather, the Maya region was controlled by numerous city states in frequently shifting tiers of power, alliances, and rivalries. There was never a single Maya power to term an "empire" that would be interested in expansion. To turn towarfs such ideas, a city state would have likely needed greater dominance and stability than any ever actually acheived. We do know there was trade contact (likely through middlemen) with the southern US, but there's no evidence for colonies or expansion.