Under the condition that the degree gap k exceeds twice the maximum multiplicity of the induced map at infinity at its periodic points, such endomorphisms have only vertical lines as periodic curves and obey the dynamical Mordell-Lang conjecture for curves.
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Local height arguments toward the dynamical Mordell-Lang conjecture
Under the condition that the degree gap k exceeds twice the maximum multiplicity of the induced map at infinity at its periodic points, such endomorphisms have only vertical lines as periodic curves and obey the dynamical Mordell-Lang conjecture for curves.