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Let 
$f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety 
$X$ defined over a number field 
$K$ and fix an initial point 
$a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety 
$V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set 
$\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.
