The quasi-integral Read-Bajraktarević functional, symmetries and Hardy-Orlicz spaces

In this talk, we prove the existence of complex local fractal functions of the Hardy-Orlicz class. A local fractal function is the fixed point of the Read-Bajraktarevic (RB) functional. The graph of the fixed point is the attractor of an appropriate iterated function system (IFS), whose construction is fairly standard. When the underlying function space is a Hardy-Orlicz space, the local fractal function of the Hardy-Orlicz class (fixed point of the RB operator) displays peculiar properties stemming from the complex conjugation. For example, it is known that both the fixed point and its graph are intrinsically real. The latter reflects as an embedding of the realified attractor of the induced iterated function system into the product [–1, 1] × [–1, 1]. The induced IFS in this case is denominated a Complex Iterated Function System (CIFS). In this talk, we provide a characterization of this type of IFSs via a quasi-integral representation of the Read-Bajraktarevic (RB) functional. The realization of a contractive IFS in the (previously untreated) analytic case is obtained via a finite-increments theorem for holomorphic functions, dating back to 1965. We prove that both the fractal function and the induced CIFS are complex extensions of their real counterparts. In a broad sense, our results generalize some of the results obtained by Massopust et al. on Lebesgue and Sobolev spaces to higher orders, dimensions, and function spaces (where a Young function now plays the role of the norm). These results somewhat show that it would be natural to extend the Read-Bajraktarević operator to other function spaces on subdomains of differentiable and real analytic manifolds. Other questions, such as the existence of fixed points of higher order, or in more sophisticated function spaces, remain open as well. Our generalizations may be useful in applications.