Structural stability of \(C^1\) diffeomorphisms. (English) Zbl 0343.58009

Robinson, Clark

J. Differ. Equations 22, 28-73 (1976).

MSC:

37C75	Stability theory for smooth dynamical systems
34D30	Structural stability and analogous concepts of solutions to ordinary differential equations
58C15	Implicit function theorems; global Newton methods on manifolds
37D99	Dynamical systems with hyperbolic behavior

References:

[1]	Abraham, R; Robbin, J, Transversal mappings and flows, (1967), Benjamin New York · Zbl 0171.44404
[2]	Anosov, D, Geodesic flows on closed Riemannian manifolds of negative curvature, (), 1969
[3]	{\scC. Camacho and R. Mañé}, Stability theorems for flows on open two dimensional manifolds.
[4]	de Melo, W, Structural stability of diffeomorphisms on two-manifolds, Invent. math., 21, 233-246, (1973) · Zbl 0291.58011
[5]	Dieudonné, J, Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201
[6]	Fenichel, N, Persistence and smoothness of invariant manifolds for flows, Indiana univ. math. J., 21, 193-226, (1971) · Zbl 0246.58015
[7]	Franks, J, Necessary conditions for stability of diffeomorphisms, Trans. amer. math. soc., 158, 301-308, (1972) · Zbl 0219.58005
[8]	{\scJ. Franks}, Notes on manifolds of Cr mappings, to appear.
[9]	Franks, J, Differentiably ω-stable diffeomorphisms, Topology, 11, 107-113, (1972) · Zbl 0232.58006
[10]	Franks, J, Time dependent stable diffeomorphism, Invent. math., 24, 163-172, (1974) · Zbl 0275.58011
[11]	Guckenheimer, J, Absolutely ω-stable diffeomorphisms, Topology, 11, 195-197, (1972) · Zbl 0246.58013
[12]	Hirsch, M; Pugh, C, Stable manifolds and hyperbolic sets, (), 133-163
[13]	{\scM. Hirsch, C. Pugh, and M. Shub}, Invariant manifolds, to appear.
[14]	Hirsch, M; Palis, J; Pugh, C; Shub, M, Neighborhoods of hyperbolic sets, Invent. math., 9, 121-134, (1970) · Zbl 0191.21701
[15]	Irwin, M, On the smoothness of the composition map, Oxford quart. J. math., 23, 113-133, (1972) · Zbl 0235.46067
[16]	Mañé, R, Expansive diffeomorphisms, (), 162-174
[17]	Moser, J, On a theorem of Anosov, J. differential equations, 5, 411-440, (1969) · Zbl 0169.42303
[18]	Nash, J, The embedding problem for Riemannian manifolds, Ann. of math., 63, 20-63, (1956) · Zbl 0070.38603
[19]	Newhouse, S, Hyperbolic limit sets, Trans. amer. math. soc., 167, 125-150, (1972) · Zbl 0239.58009
[20]	{\scS. Newhouse and J. Palis}, Cycles and bifurcation theory, to appear. · Zbl 0322.58009
[21]	Nitecki, Z, Differentiable dynamics, (1971), M.I.T. Press Cambridge, Mass
[22]	Palis, J, ()
[23]	Palis, J, On Morse-Smale dynamical systems, Topology, 8, 385-405, (1969) · Zbl 0189.23902
[24]	Palis, J; Smale, S, Structural stability theorems, (), 223-232 · Zbl 0214.50702
[25]	Peixoto, M, Structural stability on two dimensional manifolds, Topology, 1, 101-120, (1962) · Zbl 0107.07103
[26]	Pugh, C; Shub, M, The ω-stability theorems for flows, Invent. math., 11, 150-158, (1970) · Zbl 0212.29102
[27]	Robbin, J, A structural stability theorem, Ann. of math., 94, 447-493, (1971) · Zbl 0224.58005
[28]	Robinson, R.C, Structural stability of vector fields, Ann. of math., 99, 154-175, (1974) · Zbl 0275.58012
[29]	Robinson, R.C, Structural stability of C1 flows, (), 262-277
[30]	Sacker, R, A perturbation theorem for invariant manifolds and Hölder continuity, J. math. mech., 18, 705-762, (1969) · Zbl 0218.34046
[31]	Schwartz, J, Nonlinear functional analysis, (1969), Gordon and Breach New York · Zbl 0203.14501
[32]	Smale, S, Differentiable dynamical systems, Bull. amer. math. soc., 73, 747-817, (1967) · Zbl 0202.55202
[33]	Shub, M, Stability and genericity for diffeomorphisms, (), 492-514 · Zbl 0289.58013