The structure of open continuous mappings having two valences

Lyzzaik, A. K.; Stephenson, Kenneth

doi:10.1090/S0002-9947-1991-1062192-2

The structure of open continuous mappings having two valences
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by A. K. Lyzzaik and Kenneth Stephenson

Trans. Amer. Math. Soc. 327 (1991), 525-566

DOI: https://doi.org/10.1090/S0002-9947-1991-1062192-2

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Abstract:

The authors study open continuous functions which map the unit disc to compact Riemann surfaces and which assume each value in the range space (with a finite number of exceptions) either $p$ or $q$ times for some positive integers $p$, $q$. Although the questions here originated in efforts to understand mapping properties of locally univalent analytic functions, the authors remove analyticity assumptions and show that the underlying issues are topological and combinatoric in nature. The mappings are studied by embedding their image surfaces in compact covering spaces, a setting which allows the consideration of fairly general ranges and which accommodates branch and exceptional points. Known results are generalized and extended; several open questions are posed, particularly regarding the higher dimensional analogues of the results.

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Principes topologiques de la theorie des fonctions analytiques