Anosov Closing Lemma Danyu Zhang The Ohio State University April - PowerPoint PPT Presentation

Anosov Closing Lemma Danyu Zhang The Ohio State University April 24, 2019 1 / 3

Set-up Let M be a smooth manifold, U ⊂ M an open subset, f : U → M a C 1 diffeomorphism onto its image, and Λ ⊂ U a compact f -invariant set, i.e., f Λ ⊂ Λ. Definition The set Λ is called a hyperbolic set for the map f if there exists a Riemannian metric in an open neighborhood U of Λ and λ < 1 < µ such that for any point x ∈ Λ the sequence of differentials ( Df ) f n x : T f n x M → T f n +1 M , n ∈ Z , admits a ( λ, µ )-splitting, i.e., there exist x x E s / u = E s / u x M = E s n ⊕ E u decompositions T f n n such that ( Df ) f n n +1 and n � ( Df ) − 1 n +1 � ≤ µ − 1 . � � � ( Df ) f n n � ≤ λ, � f n � E u E s x x � 2 1 : T 2 → T 2 , � Example. The Arnold’s cat map on 2-torus: f = 1 1 √ √ λ = 3 − 5 , µ = 3+ 5 . 2 2 2 / 3

Anosov Closing Lemma Definition We call a sequence x 0 , x 1 , . . . , x m − 1 , x m = x 0 of points a periodic ǫ -orbit if dist( fx k , x k +1 ) < ǫ for k = 0 , . . . , m − 1. Theorem Let Λ be a hyperbolic set for f : U → M. Then there exists an open neighborhood V ⊃ Λ and C , ǫ 0 > 0 such that for ǫ < ǫ 0 and any periodic ǫ -orbit ( x 0 , . . . , x m ) ⊂ V there is a point y ∈ U such that f m y = y and dist ( f k y , x k ) < C ǫ for k = 0 , . . . , m − 1 . Reference: Introduction to the Mordern Theory of Dynamical Systems, Katok & Hasselblatt. 3 / 3