The attached link mentions the confluence of Julia set and hyperbolic functional scenario. This question is related to the classical inquiry on the stability of complex geometric transformations when partial differential equations and recursions are intensely involved. Following are the sub questions.
The theory of hyperbolic functions has developed in ways that, at first sight, does not appear any connection to hyperbolic functions. However, it does not trelate to the theory of Fibonacci numbers, an actively developing branch of modern mathematics. In 1993, the Ukrainian mathematicians Alexey Stakhov and Ivan Tkachenko developed a new approach to the theory of hyperbolic functions. Using the so-called Binet formulas, they developed a new class of hyperbolic functions called Hyperbolic Fibonacci and Lucas Functions. This idea was further developed and defined a class of symmetric Hyperbolic Fibonacci and Lucas functions.
http://www.mi.sanu.ac.rs/vismath/stakhov/
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