Homoclinic tangencies leading to robust heterodimensional cycles

We consider C (r > 1) diffeomorphisms f defined on manifolds of dimension > 3 with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism f can be C approximated by diffeomorphisms with C robust heterodimensional cycles. As an application, we show that the classic Simon-Asaoka’s examples of diffeomorphisms with C robust homoclinic tangencies also display C robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain C robust cycles after C perturbations.

[1]  Dmitry Turaev,et al.  ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS , 1996 .

[2]  D. Turaev,et al.  Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors , 2017, Nonlinearity.

[3]  C. Bonatti,et al.  ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS , 2007, Journal of the Institute of Mathematics of Jussieu.

[4]  Masayuki Asaoka.  Hyperbolic sets exhibiting ¹-persistent homoclinic tangency for higher dimensions , 2007 .

[5]  P. Berger.  Generic family with robustly infinitely many sinks , 2014, 1411.6441.

[6]  Carl P. Simon,et al.  A 3-dimensional Abraham-Smale example , 1972 .

[7]  C. Pugh.  An Improved Closing Lemma and a General Density Theorem , 1967 .

[8]  E. Pujals,et al.  Robust Transitivity in Hamiltonian Dynamics , 2011, 1108.6012.

[9]  S. Newhouse,et al.  The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms , 1979 .

[10]  Floris Takens,et al.  Bifurcations and stability of families of diffeomorphisms , 1983 .

[11]  Shlomo Sternberg,et al.  Local Contractions and a Theorem of Poincare , 1957 .

[12]  Pablo G. Barrientos.  Historic wandering domains near cycles , 2021 .

[13]  W. Kyner.  Invariant Manifolds , 1961 .

[14]  P. Alam.  ‘L’ , 2021, Composites Engineering: An A–Z Guide.

[15]  E. Pujals,et al.  Iterated Functions Systems, Blenders and Parablenders , 2015, 1603.01241.

[16]  N. Gourmelon,et al.  Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication , 2010, Ergodic Theory and Dynamical Systems.

[17]  J. Palis,et al.  High dimension diffeomorphisms displaying infinitely many periodic attractors , 1994 .

[18]  Pablo G. Barrientos,et al.  Symbolic blender-horseshoes and applications , 2012, 1211.7088.

[19]  R. Ures,et al.  New criteria for ergodicity and nonuniform hyperbolicity , 2009, 0907.4539.

[20]  Romain Dujardin.  Non density of stability for holomorphic mappings on P^k , 2016, 1610.01785.

[21]  S. Finch.  Lyapunov Exponents , 2007 .

[22]  Partial Hyperbolicity and Homoclinic Tangencies , 2011, 1103.0869.

[23]  Masayuki Asaoka.  Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension , 2021 .

[24]  A. Katok.  Lyapunov exponents, entropy and periodic orbits for diffeomorphisms , 1980 .

[25]  C. Pugh.  Against the C2 closing lemma , 1975 .

[26]  Récurrence et généricité , 2003, math/0306383.

[27]  C. Moreira.  There are no C1-stable intersections of regular Cantor sets , 2009, 0901.3131.

[28]  J. Buzzi,et al.  Measures of maximal entropy for surface diffeomorphisms , 2018, Annals of Mathematics.

[29]  Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures , 2015, 1502.06535.

[30]  Shuhei Hayashi.  Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows , 1997 .

[31]  Christian Bonatti,et al.  Persistent nonhyperbolic transitive diffeomorphisms , 1996 .

[32]  L. Díaz,et al.  Nontransverse heterodimensional cycles: stabilisation and robust tangencies , 2020, 2011.08926.

[33]  Dongchen Li.  Homoclinic bifurcations that give rise to heterodimensional cycles near a saddle-focus equilibrium , 2016, 1604.00431.

[34]  SURVEY Towards a global view of dynamical systems, for the C1-topology , 2011, Ergodic Theory and Dynamical Systems.

[35]  Pablo G. Barrientos,et al.  Robust Degenerate Unfoldings of Cycles and Tangencies , 2020 .

[36]  C. Simon.  INSTABILITY IN Diffr ( T 3 ) AND THE NONGENERICITY OF RATIONAL ZETA FUNCTIONS , 2010 .

[37]  E. Pujals,et al.  Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms , 2010, 1011.3836.

[38]  Pablo G. Barrientos,et al.  Robust tangencies of large codimension , 2017, 1707.05638.

[39]  A. Avila,et al.  C1 density of stable ergodicity , 2017, Advances in Mathematics.

[40]  Newhouse phenomenon for automorphisms of low degree in C3 , 2020 .

[41]  Pierre Berger,et al.  Generic family displaying robustly a fast growth of the number of periodic points , 2017, Acta Mathematica.

[42]  Johan Taflin.  Blenders near polynomial product maps of C , 2018 .

[43]  Homoclinic tangencies and hyperbolicity for surface diffeomorphisms , 2000, math/0005303.

[44]  T. Morrison,et al.  Dynamical Systems , 2021, Nature.

[45]  C. Bonatti,et al.  Abundance of C 1 -robust homoclinic tangencies , 2009, 0909.4062.

[46]  N. Romero.  Persistence of homoclinic tangencies in higher dimensions , 1995, Ergodic Theory and Dynamical Systems.

[47]  L. Shilnikov,et al.  On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I , 2008 .

[48]  Jacob Palis,et al.  A global view of dynamics and a conjecture on the denseness of finitude of attractors , 2018, Astérisque.

[49]  R. Ures,et al.  Abundance of hyperbolicity in the $C^1$ topology , 1995 .

[50]  Christian Bonatti,et al.  Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective , 2004 .

[51]  Robustly non-hyperbolic transitive symplectic dynamics , 2017, 1707.06473.