2022/07/28 by James David Nixon, Nixon, James David
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Fractional Differential Equations Solutions #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2208.05328
In this report we construct a family of holomorphic functions βλ,μ (s) which behave asymptotically like iterated exponentials as |s| → ∞ in the right half plane. Each βλ,μ satisfies a convenient functional relationship with nested exponentials; and has a series expansion that converges in a half-plane. They provide a nearness to the dynamics of the map eμz : ℂ→ℂ and behave asymptotically as a fractional iteration would behave. These objects are used to describe the various orbits of the exponential function. We describe where Abel equations are feasibly constructed from β. Where there exists wildly holomorphic functions with period 2 πi / λ that are holomorphic Abel functions of the form t(s+1) = eμt(s).