Anosov's closure lemma

In the theory of dynamic systems, Anosov's closure lemma states that closed pseudo-orbites of a dynamic system can be approximated by periodic orbites . It was proven by Dmitri Viktorovich Anosov .

Closure lemma

Let be a hyperbolic set of a diffeomorphism . ${\ displaystyle \ Lambda \ subset M}$ ${\ displaystyle f \ colon M \ to M}$

Then there is an open neighborhood of and positive numbers , so that for every closed pseudo-orbit of length there is one with ${\ displaystyle U}$ ${\ displaystyle \ Lambda}$ ${\ displaystyle C, \ epsilon _ {0}> 0}$ ${\ displaystyle \ epsilon <\ epsilon _ {0}}$ ${\ displaystyle \ epsilon}$ ${\ displaystyle x_ {0}, \ ldots, x_ {n-1}}$ ${\ displaystyle n}$ ${\ displaystyle y \ in M}$

{\ displaystyle f ^ {n} (y) = y}

gives with

{\ displaystyle d (f ^ {k} (y), x_ {k}) <C \ epsilon}

for .

{\ displaystyle 0 \ leq k \ leq n-1}

literature

Anatole Katok , Boris Hasselblatt : Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. ISBN 0-521-34187-6
DV Anosov, EV Zhuzhoma: Closing Lemmas, Differential Equations, Volume 48, 2012, pp. 1653–1699 (Chapter 4 Anosov Lemma, p. 1672)

Web links

Hasselblatt: Hyperbolic dynamical systems (Chapter 3.2)

Individual evidence

^ Anosov, Geodesic flows on closed Riemannian Manifolds of negative curvature, Tr. Math. Inst. Akad. Nauka SSSR, Volume 90, Moscow: Nauka 1967

[1] Anosov, Geodesic flows on closed Riemannian Manifolds of negative curvature, Tr. Math. Inst. Akad. Nauka SSSR, Volume 90, Moscow: Nauka 1967