On inverse limit stability for maps
Two not necessarily invertible maps f and g from a Banach manifold X into itself are said to be inverse limit conjugate if the shifts on the respective spaces of orbits \(\{x\in X^{{\mathbb{Z}}}| \quad f(x_ i)=x_{i+1},\quad i\in {\mathbb{Z}}\}\) and \(\{x\in X^{{\mathbb{Z}}}| \quad g(x_ i)=x_{i+1},\quad i\in {\mathbb{Z}}\}\) are conjugate in the usual sense: there exists a homeomorphism between the spaces of orbits (as subsets of \(X^{{\mathbb{Z}}}\) with the product topology), and this homeomorphism maps orbits to orbits. If f and g are homeomorphisms then inverse limit conjugacy and conjugacy are equivalent. A \(C^ r\) map, \(r\geq 1\), is said to be \(C^ r\) inverse limit stable if it lies in the \(C^ r\) interior of its equivalence class (with respect to inverse limit conjugacy). The following results are presented: A hyperbolic linear map L with skewness \(a=\max \{| L_ s|,| L_ u^{-1}| \}\) is inverse limit conjugate to Lipschitz perturbations of the form \(L+f\), provided the Lipschitz constant of f does not exceed 1-a (Theorem 2.1). Suppose a \(C^ 1\) map f has a hyperbolic invariant set I such that \(\cap_{n\in {\mathbb{N}}}f^ n(I)\) is compact. Then for all g from a \(C^ 1\) neighbourhood of f there exists an invariant set J such that f and g are inverse limit conjugate with respect to the spaces of orbits over I and J, respectively. The conjugating homeomorphism can be made arbitrarily close to the identity by choosing the neighbourhood of f sufficiently small (Theorem 3.1). From Theorem 3.1 it is deduced that ``Anosov maps'' are inverse limit stable and form an open subset of the \(C^ 1\) maps of X. Several conjectures concerning global stability, a spectral decomposition, Axiom A and Morse-Smale maps, and \(\Omega\)- inverse limit stability are formulated. Then the continuous time case (semiflows instead of maps) is discussed; analogous results are presented. Finally, applications to evolution equations are mentioned very briefly.
scientific article; zbMATH DE number 4109690
scientific article