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- W2032597913 abstract "We consider here arcs of diffeomorphisms which, being initially structurally stable, go through brusque changes (or bifurcations) in their phase portrait (space of orbits) as the parameter changes. We are mainly interested in stable arcs in the sense that nearby arcs have the same changes in their phase portraits. As we shall see, there are various natural ways of defining stability for arcs. For a of arcs starting as Morse-Smale diffeomorphisms we characterize the arcs that are stable according to the various definitions of stability. We conjecture that this large class really contains all stable arcs of diffeomorphisms starting as Morse-Smale diffeomorphisms. We present below the precise statements of the results, beginning with some definitions and known facts. The proofs will appear elsewhere. denotes a compact C°° manifold without boundary. Diff(A/) is the space of C°° diffeomorphisms of and (M) the usual C°°-topologies. For g G Diff(M), the orbit 0(x) of a point x G is defined by 0{x) = i8^(x)n G Z}; aj pointy G is called a limit point of g if for some x G and sequence np i = 1, 2, . . . , nG Z, nt —• °°, lim g'(x) = y. The closure of the limit points of g is called the limit set of g and denoted by L(g). A point x G is a periodic point of g with period n if g(x) = x and £(x) -=h x for all 0 } and {y G M dist(/(y), f{x)) —» 0 for n —• -°°} respectively. If x is a hyperbolic periodic point of g, W(x, g) and W(x, g) are smoothly immersed submanifolds of We say that a diffeomorphism g G Diff(M), with finite limit set, has an «-cycle if there is a sequence of periodic orbits 0(px), . . . , 0(pn) with O(p0) = 0(pn) and 0(pi+1) C closure (W(0(>.)) ~ °(Pi)) f o r 0 1 if no other periodic orbit can be added to the previous sequence." @default.
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- W2032597913 date "1976-01-01" @default.
- W2032597913 modified "2023-09-27" @default.
- W2032597913 title "Stable arcs of diffeomorphisms" @default.
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- W2032597913 doi "https://doi.org/10.1090/s0002-9904-1976-14073-6" @default.
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