Lacunary power series and Peano curves

Introduction. The existence of a pair of continuous functions F(t), G(t), such that the curve x F(t), y G(t) fills completely a certain square is classical. Such a curve will, as usual, be called a Peano curve. The present paper has its origin in the following question: does there exist a Peano curve x F(t), y G(t) such that the components F(t), G(t) are respectively the real and the imaginary parts of the values taken by a power series on its circle of convergence? In other words: does there exist a power series f(z) ’ a,z with radius of convergence equal to 1, continuous in the closed circle z _