Convergence of Chebyshëv continued fractions for elliptic functions

Dumas's classical theorem on the behaviour of the Chebyshëv continued fraction corresponding to an elliptic function $f(z)=\sqrt{(z-e_1)\dotsb(z-e_4)}-z^2+z{(e_1+\dotsb+e_4)}/2$ holomorphic at $z=\infty$ is extended to a fairly general class of elliptic functions. The behaviour of the Chebyshëv continued fractions corresponding to functions in that class is characterized in terms relating to the mutual position of the branch points $e_1,\dots,e_4$. The proof is based on the investigation of the properties of the solution of a certain Riemann boundary-value problem on an elliptic Riemann surface.