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The " outside " fixpoint ? - tommy1729 - 03/18/2016
Im thinking about real-entire functions , strictly increasing on the reals such that 1) they have no real fixpoint , nor at +\- oo. ( -oo is fixpoint of exp(z) + z , but x^2 does not have + oo as fixpoint although oo^2 = oo ; x^2 is not asymp to id(x) ). 2) in the univalent zone near the real axis where f maps univalent to all of C or C\y ( single value y ) , f has no fixpoint. Equivalent f^[-1](x) , the branch near the real line , has no fundamental fixpoints. So the fixpoints of f resp Inv f must lie outside the zone resp branch. I call them " outside " fixpoints. This makes me wonder about the super of f. Clearly the outside fixpoints make the super way different. For instance no fixpioints at + oo i like sexp has. Reminds me a bit of secondary fixpoint methods. Not sure what the most intresting f would be. I assume the simplest topology for the zone comes first. ; no holes for instance. Notice b^z always has " inside fixpoints " for b > eta. A theory for these outside fix would be Nice too ! Regards Tommy1729 |