Research On Several Problems Of Complex Analytical Dynamical Systems

This doctoral thesis mainly includes two aspects:1.On Lebesgue measure of polynomial Julia setsIn the field of dynamics in one complex variable,the existence of nowhere dense rational Julia sets with positive area had been a central problem until in 2012 Buff and Cheritat constructed a quadratic Julia set with positive area based on Douady's plan.They proved that there exist quadratic polynomials having Cremer Julia sets with positive area,quadratic polynomials having Siegel Ju-lia sets with positive area and infinitely renormalizable quadratic polynomials with unbounded satellite combinatorics having Julia sets with positive area.Re-cently,due to Avila,Dudko and Lyubich,it is known that for either bounded primitive combinatorics or bounded satellite combinatorics,there exist infinite-ly renormalizable quadratic polynomials having Julia sets with positive area.We observe that one can use the renormalization method to construct polyno-mials of any degree greater than 2 having Julia sets with positive area based on the existence of quadratic Julia sets with positive area.However,all of those polynomials constructed in this way are renormalizable.Therefore,a natural problem is whether there is a non-renormalizable polynomial of degree greater than 2 having a Julia set with positive area?In this thesis,we prove that there exists ??R/Z such that the cubic polynomial e2?i?(z+1)2 has a Cremer Julia set with positive area.Thus,we give a positive answer to this problem.2.The bifurcation of germs of multi-dimensional holomorphic mapsLet f be a germ of n-dimensional holomorphic map such that the origin is an isolated fixed point of each iterate of f.Then the sequence of numbers of periodic orbit hidden at the origin,(?),is well-defined.In 2006 G.Y.Zhang proved that the sequence (?) has a universal obstruction from the linear part of f at the origin,that is,the linear part of f at the origin determines which terms are zero.If the linear part of f at the origin has no obstruction,except the above universal one,we call the linear part universal.In 2014 I.Gorbovickis characterized universal linear parts for a large class of germs of holomorphic maps whose linear parts are diagonal.In this thesis we characterize universal linear parts for germs of general holomorphic maps,in particular,including all germs of holomorphic maps whose linear part are Jor-danian.For this purpose,for a germ f of general holomorphic map we establish an efficient method to count the sequense(?).