Dynamics of Holomorphic Maps: Resurgence of Fatou coordinates, and Poly-time Computability of Julia Sets

The present thesis is dedicated to two topics in Dynamics of\nHolomorphic maps. The first topic is dynamics of simple parabolic\ngerms at the origin. The second topic is Polynomial-time\nComputability of Julia sets.\\\\\n\nDynamics of simple parabolic germs. Let $F$ be a germ with a\nsimple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It\nis convenient to apply the change of coordinates $z=-1/w$ and\nconsider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The\ndynamics of a germ $f$ can be described using Fatou coordinates.\nFatou coordinates are analytic solutions of the equation\n$\\phi(f(z))=\\phi(z)+1.$ This equation has a formal solution\n \\[\\tilde\\phi(z)=\\text{const}+z+A\\log z+\\sum_{j=1}^\\infty b_jz^{-j},\\] where\n $\\sum b_jz^{-j}$ is a divergent power series. Using \\'Ecalle's Resurgence Theory we show\nthat $\\tilde{\\phi}$ can be interpreted as the asymptotic expansion of\nthe Fatou coordinates at infinity. Moreover, the Fatou coordinates\ncan be obtained from $\\tilde \\phi$ using Borel-Laplace\nsummation. J.~\\'Ecalle and S.~Voronin independently constructed a\ncomplete set of invariants of analytic conjugacy classes of germs\nwith a parabolic fixed point. We give a new proof of validity of\n\\'Ecalle's construction.\n\\\\\n Computability of Julia sets. Informally, a compact subset of\nthe complex plane is called \\emph{computable} if it can be\nvisualized on a computer screen with an arbitrarily high precision.\nOne of the natural open questions of computational complexity of\nJulia sets is how large is the class of rational functions (in a\nsense of Lebesgue measure on the parameter space) whose Julia set\ncan be computed in a polynomial time. The main result of Chapter II\nis the following: Theorem. Let $f$ be a rational\nfunction of degree $d\\ge 2$. Assume that for each critical\npoint $c\\in J_f$ the $\\omega$-limit set $\\omega(c)$ does not contain\neither a critical point or a parabolic periodic point of $f$. Then\nthe Julia set $J_f$ is computable in a polynomial time.

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