In　this　paper,　by　introducing　the　concept　of　topological　equivalence　on　measure　chain,　we　investigate　the　relationship　between　the　linear　system　x(Delta)　=　A(t)x　and　the　nonlinear　system　x(Delta)　=　A(t)x　+　f(t,x).　Some　sufficient　conditions　are　obtained　to　guarantee　the　existence　of　a　equivalent　function　H(t,x)　sending　the　(c,　d)-quasi　bounded　solutions　of　nonlinear　system　x(Delta)　=　A(t)x　+　f(t,x)　onto　those　of　linear　system　x(Delta)　=　A(t)x.　Our　results　generalize　the　Palmer＇s　linearization　theorem　in　[K.J.　Palmer,　A　generalization　of　Hartman＇s　linearization　theorem,　1.　Math.　Anal.　Appl.　41　(1973)　753-758]　to　dynamic　equation　measure　chains.　In　the　present　paper,　we　give　a　new　analytical　method　to　study　the　topological　equivalence　problem　on　measure　chains.　As　we　will　see,　due　to　the　completely　different　method　to　investigate　the　topological　equivalence　problem,　we　have　a　considerably　different　result　from　that　in　the　pioneering　work　of　Hilger　[S.　Hilger,　Generalized　theorem　of　Hartman-Grobman　on　measure　chains,　J.　Aust.　Math.　Soc.　Ser.　A　60　(2)　(1996)　157-191].　Moreover,　we　prove　that　equivalent　function　H(t,x)　is　also　omega-periodic　when　the　systems　are　omega-periodic.　Hilger　[S.　Hilger,　Generalized　theorem　of　Hartman-Grobman　on　measure　chains,　J.　Aust.　Math.　Soc.　Ser.　A　60　(2)　(1996)　157-191]　never　considered　this　important　property　of　the　equivalent　function　H　(t,　x).　(c)　2007　Elsevier　Inc.　All　rights　reserved.