M Dwarf Planet Habitability

The habitability of M dwarf planets has been debated greatly, as their parent stars possess both beneficial and detrimental qualities for the development of life. Initially, the astrobiological community questioned their habitability (Dole 1964), but as research and modeling techniques have improved, astrobiologists have become more accepting of the idea of life on M dwarf planets (Shields et al. 2016). The question of these planets’ habitability has great significance, because their long lifespans and commonality in the universe make them legitimate candidates for a plethora of extrasolar spacecraft missions, and potentially for the first discovery of life in other systems

Fixed points of composition sum operators

In the renormalization analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the form fn(z) = ℓΣi=1ai(z)fni(αi(z)), where the αi are affine contractions and each ni is either n - 1 or n - 2. We develop a general theory of these fixed points by regarding them as fixed points of ‘composition sum operators’, and apply this theory to test for fixed points in classes of complex analytic functions with various key types of singularities. Finally we demonstrate the construction of the full space of fixed points of one important class, arising from the much studied operator M defined by Mf(z) = f(-ωz) + f(ω2z + ω), ω = (√5 - 1)/2. The construction reveals previously unknown solutions