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SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY

Communicación en el III Congreso Latinoamericano de Matemáticos August 31th.-September 4th. 2009. SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY. Eleonora Catsigeras Marcelo Cerminara Heber Enrich. Instituto de Matemática. Facultad de Ingeniería.

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SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY

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  1. Communicación en el III Congreso Latinoamericano de Matemáticos August 31th.-September 4th. 2009 SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY Eleonora Catsigeras Marcelo Cerminara Heber Enrich Instituto de Matemática. Facultad de Ingeniería. Universidad de la República URUGUAY eleonora@fing.edu.uy

  2. ASSUMPTIONS: • M is a 2-dimensional compact manifold • has a dissipative saddle periodic point with a homoclinic quadratic tangency • is a one-parameter family generically unfolding the homoclinic tangency Theorems of Newhouse 1970, 1974, 1979 and Robinson 1983:

  3. Theorems of Newhouse 1970, 1974, 1979 and Robinson 1983: OPEN QUESTION: HAVE THE INFINITELY MANY COEXISTING SINKS SIMULTANEOUS CONTINUATIONS IN SOMEOPEN NEIGHBORHOOD IN THE FUNCTIONAL SPACE? ?

  4. THEOREM 1: There exists an open interval of parameter values ,and a residual subset such that: For all 1. exhibits infinitely many coexisting sinks (Newhouse-Robinson) 2. There exists a infinite-dimensional manifold such that:

  5. THEOREM 2:

  6. Route of the proofs of Theorems 1 and 2 In SIX STEPS: STEP 1: Apply Newhouse Theorem:

  7. STEP 2: • Consider TRIVIALIZING COORDINATES OF THE INVARIANT FOLIATIONS in the neighborhood • Use dissipative hypothesis of the given hyperbolic periodic point exhibiting a quadratic homoclinic tangency in f sub0, to obtain STRONG DISSIPATIVE CONDITION of the hyperbolic set for all • Apply r-normality to conclude that • The STABLE LOCAL FOLIATION is • while the unstable foliation is not necessarily more than

  8. STEP 3: Compute the iteration of f in the trivializing coordinates as in [PT 1993] (up to non trivial adaptations, using C3 differentiability), to conclude the following : LEMMA (Uniform approximation to the quadraticfamily):

  9. LEMMA (Uniform approximation to the quadraticfamily):

  10. STEP 4 Consequences of Lemma:

  11. STEP 4 Consequences of Lemma:

  12. STEP 5:

  13. STEP 5:

  14. STEP 6 (CONCLUSIONS): End of the proof of Theorem 1:

  15. STEP 6 (CONCLUSIONS): End of the proof of Theorem 1:

  16. STEP 6 (CONCLUSIONS):

  17. STEP 6 (CONCLUSIONS): End of the proof of Theorem 2:

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