Partial hyperbolicity far from homoclinic bifurcations

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387

Partial Hyperbolicity and Homoclinic Tangencies

We show that any diffeomorphism of a compact manifold can be C1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure

Invariance Entropy of Hyperbolic Control Sets

In this paper, we improve the known estimates for the invariance entropy of a nonlinear control system. For sets of complete approximate controllability we derive an upper bound in terms of Lyapunov exponents and for uniformly hyperbolic sets we obtain a similar lower bound. Both estimates can be applied to hyperbolic chain control sets, and we prove that under mild assumptions they can be merged into a formula

A (short) survey on Dominated Splitting

We present here the concept of Dominated Splitting and give an account of some important results on its dynamics.Comment: 19 page

Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms

We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction)

On Loops in the Hyperbolic Locus of the Complex H\'enon Map and Their Monodromies

We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex H\'enon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real H\'enon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex H\'enon map.Comment: 17 pages, 9 figures. For supplemental materials, see http://www.math.kyoto-u.ac.jp/~arai