OntheqcrigidityofrealpolynomialsWeixiaoShenGradnateSchoolofMathematicalSciences,UniversityofTokyo6th,2000TheFatouconjecture(ortheHDconjecture):binatorialrationalfunctionsarequasiconformallyconjugate(thisstate-binatorialrigidityconjecture);andarationalmapotherthanaLatt\‘esexample,carriesnoinvariantline?eldontheJuliaset(thisisnamedthequasiconformalrigidityconjecture,ortheNILFconjecturefollowingMc-MullenandSullivan).In[18],?eld(NILF)’,besideasolutiontoNILFconjecture,-tionwasonlydoneforpolynomialswithonlyonecriticalpointinitsJuliaset,totheatlthor’?nitelyrenormalizable-orwhichisreal,therecannotbeinvariantline?eldsup-ozand,see$[5],[16]$.ItisalsoprovedrecentlybyLevinandvanStrienthatthenoinvariantline?eldconjectureholdsforrealpolynomialwhichhasonlyacriticalpoint,see[9],[10].In[6]and[12],ase;morerecently,Shishikura[23]-escapingcriticalpointandcannotberenormalized(in?nitelymanytimes)toaquadraticpolynomial,see$[2],[3]$Inthisresult,thepolynomialisallowedtohaveseveralcriticalpoints,[21],theNILFconjectureisprovedforallrealpolynomialswhoserealcriticalpointsarealltu
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