On stochastic sea of the standard map

Abstract

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.

36 pages, 5 figures

References in corpus (2)

• Bers and Hénon, Painlevé and Schroedinger
• Some remarks on the geometry of the Standard Map

Cited by in corpus (15)

• Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems
• Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
• Eigenstate entanglement between quantum chaotic subsystems: universal transitions and power laws in the entanglement spectrum
• Ordered level spacing probability densities
• Statistical characterization of discrete conservative systems: The web map
• Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux
• Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
• Steady Euler flows on ${\mathbb{R}}^3$ with wild and universal dynamics
• Bounds on the Lyapunov exponent via crude estimates on the density of states
• Persistent massive attractors of smooth maps
• Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems
• Lyapunov exponents for random perturbations of coupled standard maps
• Mixed Dynamics in a Parabolic Standard Map
• Diffusion limit for a slow-fast standard map
• Hyperbolicity and Abundance of Elliptical Islands in Annular Billiards