Homeomorphic conjugacy of automorphisms on the torus

Abstract

F(x) = L(x) + P(x) + c where P: Rn-+Rn is a continuous periodic map (i.e., P(x+v) =P(x), xERn, vEZn) satisfying P(0) =0. This fact is established by considering P(x) = F(x) -L(x) -c. Clearly P(0) = 0. In view of the fact that LvEZn whenever vEZn we have ir(P(x+v) — P(x)) =w(F(x+v) — F(x) — Lv) = y(wx + irv) —y(wx) — wLv = 7(7^) — 7(7^) = 0 ; consequently P(x+v)—P(x) is always in Zn. Since Rn is connected and Zn is discrete P(x+v)—P(x) is (for v fixed) independent of x. Thus P(x + v) - P(x) = P(0 + v) - P(0) = P(v) = F(v) - Lv - c = G(v) —Lv which is zero by definition Zn = G Zn. We shall call L the linear part, P the periodic part, and c the constant part of the lifting F and when necessary place subscripts on these symbols to indicate the mapping on Tn from which they came. The linear, periodic, and constant part of a lifting are unique; for let L', P', c', be other ones. F(x) =L(x)+P(x)+c = L'(x)+P'(x)+c', yields L(x)-L'(x) =P(x)-P'(x)+c-c'. The right-hand side of the

Locations

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