Existence of periodically invariant curves in 3-dimensional measure-preserving mappings

Abstract

With the aid of a normal form of a family of measure-preserving mappings in dimension 3, which is deduced in this paper, we prove that there are periodically invariant curves which survive the nonlinear perturbations in the generic case.

Existence of invariant tori in three-dimensional measure-preserving mappings

“…To simplify the situation without loss of generality, we assume fo(z) = z, g'~(z) >1 Cl > 0 (z e [1,2]), here g~ > 0 is called the second twist condition. The role it plays is similar to the twist condition in the area-preserving mappings.…”

“…where g*(cO, O) = O, co ~ So, c [1,2]. So, is a Cantor set with positive Lebesgue measure which tends to 1 as do tends to O.…”

“…To make the problem easy to state, we study the following mapping in the complex domain: if the o9 ~ [ 1,2] satisfies the infinitely many inequalities of (4) with Ko in the place of K. The mapping m l (o9) = 9Jo a (o9)Mo(o9)9.io(o9) defined in a smaller domain Dx: Jim (p[ ~< p, [Im Vii ~< P]ql < a, is given by { ~oi = ~ + o9 + rt + ~(p, 0, rt, o9), @i @+go(rl, og)+gx(rl, og) + W(tP, qJ, q, og), (7) q l q + H(tp, ~b, q, o9), where U, V and W satisfy the equations: f u(~o + o9. To make the problem easy to state, we study the following mapping in the complex domain: if the o9 ~ [ 1,2] satisfies the infinitely many inequalities of (4) with Ko in the place of K. The mapping m l (o9) = 9Jo a (o9)Mo(o9)9.io(o9) defined in a smaller domain Dx: Jim (p[ ~< p, [Im Vii ~< P]ql < a, is given by { ~oi = ~ + o9 + rt + ~(p, 0, rt, o9), @i @+go(rl, og)+gx(rl, og) + W(tP, qJ, q, og), (7) q l q + H(tp, ~b, q, o9), where U, V and W satisfy the equations: f u(~o + o9.…”

“…Fortunately, we have the Lemmas (3), (4) which state that Xj(0, co) ~)(0, co) can be extended to [1,2] There may be isolated points in the intersection of Cantor sets, but its Lebesgue measure is zero. However, the extension method for gl(0, co) is useless for gj (0, co) (j/> 2) since the corresponding :~j (0, co) and ~ (0, co) are only defined on ~j (co).…”

“…Although 01 (0, o,) is only defined on the set ~o(o,), it can be extended to the interval [1, 2] by (9) in a natural way, since Xo(t/, co) and ~'o(r/, co), replacing 27(t/, co) and ~'(ff, co) in(9), are well defined for co ranging on the interval[1,2]. (09) with ~n(CO) ~--~n-1 (O,), By Lemma (1), which is proved in the next section, there exist a Cantor set So(o,) = ~o(o,).…”