User luka thaler - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:04:01Z http://mathoverflow.net/feeds/user/22547 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84742/how-is-the-julia-set-of-fg-related-to-the-julia-set-of-gf/92763#92763 Answer by Luka Thaler for How is the Julia set of $fg$ related to the Julia set of $gf$? Luka Thaler 2012-03-31T17:02:57Z 2012-03-31T17:02:57Z <p>Hy I'm not shure that your restriction map $f: J(gf)\rightarrow J(fg)$ is right. Insted of looking at Julia set it is equivalent to look at Fatou set where is more natural work with mapings. Since you've put $f:{(gf)^n}_n \rightarrow {(fg)^n}_n$ it clearly does not work. Insted you have to use conjugation $f\circ-\circ f^{-1}$, since you have rational function of deg>1 the inverse is not properly defined. So in the neighbour of the periodic points of $gf$ which are not critical (you dont have inverse) for $f$ you get same dynamics. </p> <p>As you know fixed points play important role in this theory so you can try to compare fixed points (or periodic) of $fg$ and $gf$.</p> <p>I think there are some results if you have $g$ an $f$ that commutes and than you compare dimaics of $f$ and $g$ with dynamics of $fg$.</p>