Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Fractal dimensions and infinitely many attractors.

*(English)*Zbl 0790.58014
Cambridge Studies in Advanced Mathematics. 35. Cambridge: Cambridge University Press. x, 234 p. (1993).

For \(p\) a saddle fixed point for a diffeomorphism \(\varphi\) of a surface one considers the situation where stable and unstable manifolds \(W^ s(p)\) and \(W^ u(p)\) intersect each other. Each point of the intersection is called homoclinic. Already if the intersection is transversal there is a surprising complexity of the dynamics: there are so-called Smale horseshoes. In the late sixties S. Newhouse started to investigate bifurcations from a homoclinic tangency (\(W^ u(p)\) tangent to \(W^ s(p)\)) at a point, which lead to cascades of phenomena which amazing complexity far overpasses the complexity of hyperbolic “horseshoe” invariant sets. These phenomena are investigated in the reviewed monograph.

The book is devoted mainly to proofs of the following theorems:

1. For a generic 1-parameter family of diffeomorphisms \(\varphi_ \mu\) with \(\varphi_ 0 = \varphi\) having a homoclinic tangency, there exist intervals of parameters converging to \(\mu = 0\) so that \(\varphi_ \mu\) exhibits a tangency between stable and unstable manifolds of some saddle periodic orbits for a dense set \(R\) of parameters in these intervals. Assuming \(\text{det }D\varphi(p) < 1\) there are infinitely many sinks for each \(\varphi_ \mu\) for \(\mu\) from a residual set \(R\).

The proof goes along, and modifies, a theory by S. Newhouse. The idea is to use cascades of transversal homoclinic points, hence cascades of invariant hyperbolic sets, as well as cascades of secondary homoclinic tangencies.

To know that Cantor sets of stable and unstable leaves of hyperbolic sets persistently intersect the authors use the notion of thickness, a numerical invariant of a Cantor set. The Cantor set of stable leaves of a hyperbolic set has an arbitrarily large thickness because it is a perturbation of an invariant Cantor set in the interval for \(g_{\nu(\mu)}(y) = y^ 2 - \nu(\mu)\) (such families are in the limit of \(\varphi_ \mu^{n(\mu)}\) for \(\mu\to 0\) and for respective blowings up in neighbourhoods of the tangency point and in parameters in a neighbourhood of \(\mu = 0)\).

This Theorem disproves R. Thom’s conjecture that generically there is only a finite number of attractors.

Also other phenomena are described in the book: homoclinic tangencies yield cascades of critical saddle-node bifurcations, which yield Hénon- like attractors.

2. (This is opposite to 1. and discusses the prevalence of the parameters with non-intersecting Cantor sets of stable and unstable leaves.) If \(\varphi_ 0\) has an invariant hyperbolic set \(\Lambda\) containing a saddle \(p\) which exhibits a homoclinic tangency \(q\) then for the maximal invariant set \(\Gamma_ \mu\) in a neighbourhood of the orbits of \(p\) and \(q\), for the bifurcation set \(B: = \{\mu: \Gamma_ \mu \text{ is not hyperbolic}\}\) if Hausdorff dimension HD\((\Lambda) < 1\) then \[ {\text{Leb}[(B) \cap (0,\mu_ 0)]\over \mu_ 0} \to 0 \quad\text{as}\quad \mu_ 0 \to 0. \] The detailed exposition and proof is in Ann. Math., II. Ser. 125, 337-374 (1987; Zbl 0641.58029), the paper included in the book as Appendix. The authors mention a converse theorem where \(\text{HD}(\Lambda) > 1\) [the first author and J.-Ch. Yoccoz, ‘Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension’, to appear]. A concrete situation to which this is applied is Smale’s Axiom A without circles for \(\mu < 0\) and a cycle for \(\mu = 0\).

The book is quite advanced, describes new ideas and directions in smooth dynamics. On the other hand it contains a lot of a background material for a non-specialist reader.

Related is the reviewer’s paper in Bol. Soc. Bras. Mat. 18, No. 1, 29-79 (1987; Zbl 0759.58034).

The book is devoted mainly to proofs of the following theorems:

1. For a generic 1-parameter family of diffeomorphisms \(\varphi_ \mu\) with \(\varphi_ 0 = \varphi\) having a homoclinic tangency, there exist intervals of parameters converging to \(\mu = 0\) so that \(\varphi_ \mu\) exhibits a tangency between stable and unstable manifolds of some saddle periodic orbits for a dense set \(R\) of parameters in these intervals. Assuming \(\text{det }D\varphi(p) < 1\) there are infinitely many sinks for each \(\varphi_ \mu\) for \(\mu\) from a residual set \(R\).

The proof goes along, and modifies, a theory by S. Newhouse. The idea is to use cascades of transversal homoclinic points, hence cascades of invariant hyperbolic sets, as well as cascades of secondary homoclinic tangencies.

To know that Cantor sets of stable and unstable leaves of hyperbolic sets persistently intersect the authors use the notion of thickness, a numerical invariant of a Cantor set. The Cantor set of stable leaves of a hyperbolic set has an arbitrarily large thickness because it is a perturbation of an invariant Cantor set in the interval for \(g_{\nu(\mu)}(y) = y^ 2 - \nu(\mu)\) (such families are in the limit of \(\varphi_ \mu^{n(\mu)}\) for \(\mu\to 0\) and for respective blowings up in neighbourhoods of the tangency point and in parameters in a neighbourhood of \(\mu = 0)\).

This Theorem disproves R. Thom’s conjecture that generically there is only a finite number of attractors.

Also other phenomena are described in the book: homoclinic tangencies yield cascades of critical saddle-node bifurcations, which yield Hénon- like attractors.

2. (This is opposite to 1. and discusses the prevalence of the parameters with non-intersecting Cantor sets of stable and unstable leaves.) If \(\varphi_ 0\) has an invariant hyperbolic set \(\Lambda\) containing a saddle \(p\) which exhibits a homoclinic tangency \(q\) then for the maximal invariant set \(\Gamma_ \mu\) in a neighbourhood of the orbits of \(p\) and \(q\), for the bifurcation set \(B: = \{\mu: \Gamma_ \mu \text{ is not hyperbolic}\}\) if Hausdorff dimension HD\((\Lambda) < 1\) then \[ {\text{Leb}[(B) \cap (0,\mu_ 0)]\over \mu_ 0} \to 0 \quad\text{as}\quad \mu_ 0 \to 0. \] The detailed exposition and proof is in Ann. Math., II. Ser. 125, 337-374 (1987; Zbl 0641.58029), the paper included in the book as Appendix. The authors mention a converse theorem where \(\text{HD}(\Lambda) > 1\) [the first author and J.-Ch. Yoccoz, ‘Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension’, to appear]. A concrete situation to which this is applied is Smale’s Axiom A without circles for \(\mu < 0\) and a cycle for \(\mu = 0\).

The book is quite advanced, describes new ideas and directions in smooth dynamics. On the other hand it contains a lot of a background material for a non-specialist reader.

Related is the reviewer’s paper in Bol. Soc. Bras. Mat. 18, No. 1, 29-79 (1987; Zbl 0759.58034).

Reviewer: F.Przytycki (Warszawa)

##### MSC:

37Cxx | Smooth dynamical systems: general theory |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |