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Control-12-03-03-for-web

Control-12-03-03-for-web - New Phenomena Associated With...

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by S. Newhouse Michigan State University 1
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Summary: 1. Def of homoclinic points, basic properties 2. the homoclinic relation: on hyperbolic sets 3. generic consequences of homoclinic tangencies (a) C r , r > 1 , homoclinic set (h-closure) Closure ( sinks ) Closure ( sources ) ; homoclinic set has Hausdorff dim = 2, no SRB measure, no prin- cipal symbolic extension (b) C 1 area preserving: elliptic periodic points dense; no symbolic exten- sions at all; homoclinic set has Hausdorff dim = 2. 2
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M = compact surface D r ( M ) = space of C r diffeomorphisms of M with the uniform C r topology, r 1 . D r ω ( M ) = space of volume preserving diffeomorphisms Let f ∈ D r ( M ) . Λ = Λ( f ) be a compact invariant hyperbolic basic set That is, f (Λ) = Λ , neighborhood U Λ with n f n ( U ) = Λ f | Λ is topologically transitive , periodic points dense splitting TM Λ = E u E s and constants C > 0 , λ > 1 such that | Df n x | E s x | ≤ - n , | Df - n x | E u x | ≤ - n for n 0 , x Λ 3
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Properties of hyperbolic basic sets: persistence and conjugate stability: For g C 1 near f , n g n ( U ) = Λ( g ) is a hyperbolic basic set a homeomorphism h : Λ( g ) Λ( f ) such that fh = hg . invariant manifolds: for x Λ( f ) , W u ( x ) = { y M : d ( f - n y, f - n x ) 0 , n → ∞} W s ( x ) = { y M : d ( f n y, f n x ) 0 , n → ∞} are injectively immersed C r curves in M , depending continuously on compact parts 4
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Definitions ˆ W u ( x ) = W u ( x ) \ { x } , ˆ W s ( x ) = W s ( x ) \ { x } Homoclinic Point : q ˆ W u ( x ) ˆ W s ( y ) , for x, y Λ , hyperbolic basic set Transverse Homoclinic Point: ˆ W u ( x ) q ˆ W s ( y ) Homoclinic Tangency: ˆ W u ( x ) tangent to ˆ W s ( y ) at q Homoclinic Relation (h-relation): Λ 1 Λ 2 if x Λ 1 , y Λ 2 , q ˆ W u ( x ) ˆ W s ( y ) , transverse, and vice-versa Fact:
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