Pseudo orbit

In the theory of dynamic systems, a pseudo-orbit of an iteration is a sequence of points, each of which approximates the pixel of the previous point.

When modeling dynamic systems on the computer, you usually only get a pseudo orbit and not an exact orbit because of the unavoidable rounding errors.

Definitions

Let it be a representation of a metric space on itself. ${\ displaystyle f \ colon X \ to X}$

One consequence is a -pseudo-orbit for one if for all the inequality ${\ displaystyle \ left \ {x_ {i} \ right \} _ {i \ in \ mathbb {N}}}$ ${\ displaystyle \ epsilon}$ ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle i \ geq 0}$

{\ displaystyle d (x_ {i + 1}, f (x_ {i})) <\ epsilon}

applies.

A sequence is a closed or periodic pseudo-orbit of length for one if for all ${\ displaystyle \ left \ {x_ {i} \ right \} _ {0 \ leq i \ leq n-1}}$ ${\ displaystyle \ epsilon}$ ${\ displaystyle n}$ ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle 0 \ leq i \ leq n-2}$

{\ displaystyle d (x_ {i + 1}, f (x_ {i})) <\ epsilon}

and also the inequality

{\ displaystyle d (x_ {0}, f (x_ {n-1})) <\ epsilon}

applies.

Shading

In general, a pseudo orbit does not have to be approximated by real orbites. However, this is possible under certain conditions ( shading lemma ), especially for pseudo-orbites in hyperbolic sets .

Anosov's closure lemma makes a corresponding statement for periodic pseudo-orbites .

literature

Dmitri Anossow : Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder American Mathematical Society, Providence, RI 1969
Rufus Bowen : On Axiom A diffeomorphisms. Regional Conference Series in Mathematics, No. 35. American Mathematical Society, Providence, RI, 1978. ISBN 0-8218-1685-3
Rufus Bowen: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Second revised edition. With a preface by David Ruelle. Edited by Jean-René Chazottes. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008. ISBN 978-3-540-77605-5
Charles Conley : The gradient structure of a flow. I. With a comment by R. Moeckel. Ergodic Theory Dynam. Systems 8 ^* (1988), Charles Conley Memorial Issue, 11-26, 9.

Web links

Gunesch: Introduction to dynamic systems