On a Rigidity Problem of Beardon and Minda

In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let $$E\not \subset {\hat{\mathbb {R}}}$$ E ⊄ R ^ be an arc or a circle. If a map $$f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}$$ f : C ^ ↦ C ^ preserves cross-ratios in E, then f is a Möbius transformation when $${\bar{E}}\not =E$$ E ¯ ≠ E and f is a Möbius or conjugate Möbius transformation when $${\bar{E}}=E$$ E ¯ = E , where $${\bar{E}}=\{{\bar{z}}|z\in E\}$$ E ¯ = { z ¯ | z ∈ E } . (II) Let $$E\subset {\hat{\mathbb {R}}}$$ E ⊂ R ^ be an arc satisfying the condition that the cardinal number $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 . If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 is necessary.