Let X be a modular curve - for example, the λ-line - and let α:X → X be an automorphism, say, that doesn't preserve the cusps. How does α affect the general Hecke orbit structure of X? In certain contexts - thanks to the work of a number of people in Model Theory - finiteness (of the intersection of one Hecke orbit with the image under α of another Hecke orbit) is known. I will describe a few of those known results, and formulate some questions about bounds connected to them.