Topological classification of unitary functions of arbitrary genus (Q2759267)

Let \(P\) be a compact Riemann surface of genus \(g\), \(\tau\) an antiholomorphic involution of \(P\), and \(f\) a meromorphic function on \(P\). Then a triple \((P,\tau ,f)\) is called a unitary function if \(f(\tau(z))=(\overline{f(z)})^{-1}\). In this paper the authors give a necessary and sufficient condition for the set of unitary functions with a given topological type to be non-empty. The topological type of a unitary function on a non-separating real curve is defined as follows. Let \(P^{\tau}\) be the set of fixed points of \(\tau\). The elements of \(P^{\tau}\) are pairwise disjoint simple closed contours and called ovals. A pair \((P,\tau)\), which is a real algebraic curve, is said to be separating (resp. non-separating) if the complement of \(P^{\tau}\) in \(P\) is disconnected (resp. connected). The topological type of \((P,\tau)\) is a triple \((g,k,\varepsilon)\), where \(k\) is the number of ovals in \(P^{\tau}\), and \(\varepsilon\) is equal to 1 (resp. 0) if \((P,\tau)\) is separating (resp. non-separating). The topological type of the pole divisor of \(f\) is the sequence of numbers \(A=(a_1, \ldots, a_m)\), where \(a_i\) is the order of pole at \(p_i\in P\). If \(g>0\), then the characteristic of \(f\) of degree \(n\) with simple critical values (except 0 and \(\infty\)) is defined to be the factor \(r\) \((1\leq r <n)\) of \(n\) such that the monodromy group of \(f\) is generated by the permutations \((1,\ldots ,n)\) and \((1,r+1)\). Put \(r=1\) if \(g=0\). The index of \((P,\tau,f)\) on an oval \(c\) in \(P^{\tau}\) is defined to be the rotation number of \(f(p)\) around zero when \(p\) makes one revolution around \(c\). Then the topological type of a unitary function on a non-separating real curve is the tuple \((g,n,0| I| A,r)\), where \((g,k,0)\) is the topological type of \((P,\tau)\), \(n\) is the number of sheets of \(f\), \(I=(i_1,\ldots ,i_k)\) is the sequence of the indices of \((P,\tau,f)\), \(A\) is the topological type of the pole divisor of \(f\), and \(r\) is the characteristic of a meromorphic function close to \(f\) with the same topological type of divisors and with simple critical values (except 0 and \(\infty\)). The topological type of a unitary function on a separating real curve is also considered.