Let $X$ be a complete metric space and a continuous map $ f: X \to X $. A condition sufficient for a ball $ B(a, r) \subseteq X $ to be invariant by $ f $, that is $ f \left (B (a, r) \right) \subseteq B (a, r) $, is that:
- $ f|_{B(a,r)} $ is a $\lambda$-contraction,
- $ d(f (a), a) <r (1-\lambda)$.
Are there other criteria over $f$ for invariance of ball $ B (a, r) $? That is, not based on these two conditions over $f$ ? Any references?