The Homoclinic Birkhoff "Remarkable Curves" in Euclidean Spaces
by
Dmitry W. Serow
St.Petersburg Nuclear Physics Institute Russian Academy of Science

In 1932, Birkhoff discovered his ``remarkable curve'', which is actually a strange set and not an arc or topological circle. He constructed a map on an annulus with an unusual invariant set \Gamma (Birkhoff curve). His curve is the boundary set for the region G⁽⁰⁾ which contains one boundary circle. The curve is also the boundary set for the region G⁽¹⁾ which contains the other boundary circle.

Recently I stutied a dynamical and topological structure of the Birkhoff curve in the dissipative homoclinic situation on the plane: for an action \psi_k in Aut (E²) the Birkhoff curve is either indecomposable continuum or the union of two indecomposable continua, and furthemore it is a compact being common boundary of the infinite number of regions.

It is clear that the Birkhoff curves there subsist in n-dimensional Eucliadean spaces.

Theorem Let \psi_k in Aut (Eⁿ) be a dissipative action on Eⁿ, and all the fixed p_\nu, # p = m of \psi are saddle one whose stable W^s(p_\nu) and unstable W^u(p_\nu) manifolds are transversal, and Clos( \cup _{\nu <= m} W^u(p_\nu)) is connected. Then the following statements are equivalent:

(1) there exists the Birkhoff curve \Upsilon with respect to the action \psi;

(2) (W^s(p_\nu) \cap W^u(p_\nu)) \p_\nu =/= Ø for all \nu <= m.

Moreover than the Birkhoff curve \Upsilon is the common boundary of infinite number of regions G⁽ⁱ⁾, 0 <= i <= \infty, and moreover rotation numbers \alpha(G^(\mu)) and \alpha(G^(\nu)) for any two regions G^(\mu) and G^(\nu), \mu =/= \nu, are irrational and different.

[1] D. W. Serow The Homoclinic Birchoff Curves in the Plane, Nonlinear Dynamical Systems, 2, St. Petersburg University Press,  1999, pp. 194 - 203. (in Russian)

Date received: February 15, 2000