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# MAT335 Problem Set 5 Self Generated Solutions.pdf

5 Pages
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Department
Mathematics
Course
MAT335H1
Professor
Dietrich Burbulla
Semester
Summer

Description
SolutionstoMAT335H1FProblemSet5Chapter1112478Chapter121a1c1e4578Chapter131c1d6843Ch111Yes4823and5627so56isbefore48intheSarkovskiiorderingBytheTheoremonpage138itispossibleforFtohaveaperiodicpointofperiod48butnotoneofperiod5645Ch112No1762119623so176isbefore96intheSarkovskiiorderingBySarkovskiisTheoremifFhasaperiodicpointoforder176itmustalsohaveaperiodicpointofperiod96Ch114Bothfunctionsbelowhave4cyclesTheoneonthelefthascyclesofallotherperiodstheoneontherighthasonlycyclesofperiods12and433BBBB22BABABABA11BABABABBA01230123Forthegraphontheleftthe4cycleis01230Thisgraphhascyclesofallotherperiodsbecauseithasa3cycleonehasbeendrawninTheequationofthegraphisx1if0x2Fx3x9if2x3andsoF3474F74114F114343AlternativelyyoucanuseObservation1frompage134F0103013soFhasaxedpointin01whichisapointofperiod3forFForthegraphontherightone4cycleis03120sotwo2cyclesare01and32Weclaimithasnocyclesofanyotherperiodsbecausewe4shallshowithasno8cycleLetGbethefunctionrstcheckthatGxxforallx0123whichmeanseverypointin0123isona4cycleora2cycleAtypical4cycleisshownonthegraphSoifGhasapointwithperiod8itmust8beintheinterval12Supposex12andGxxIfGx0123thentheorbitofxunderGleavestheinterval12forgoodsinceG0123andG2301
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