Takens' last problem and existence of non-trivial wandering domains

Abstract

In this paper, we give an answer to a $C^r$ $(2\le r<\infty )$ version of the open problem of Takens in [Nonlinearity, 21 (2008), no.3, T33-T36] which is related to historic behavior of dynamical systems. To obtain the answer, we show the existence of non-trivial wandering domains near a homoclinic tangency, which is conjectured by Colli-Vargas [Ergod. Th. & Dynam. Sys., 21 (2001), 1657-1681]. Concretely speaking, it is proved that any Newhouse open set in the space of $C^r$-diffeomorphisms on a closed surface is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior. Moreover, this result implies an answer in the $C^r$ category to one of the open problems of van Strien [Discrete Conti. Dynam. Sys., 27 (2010), no.2, 557-588] which is concerned with wandering domains for Hénon family.

References in corpus (1)

• Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies

Cited by in corpus (30)

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• Geometric Lorenz flows with historic behavior
• Emergence via non-existence of averages
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• Historic behaviour for nonautonomous contraction mappings
• Non-trivial wandering domains for heterodimensional cycles
• Complete set of invariants for a Bykov attractor
• Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps
• Abundance of observable Lyapunov irregular sets
• Historic wandering domains near cycles
• Highly irregular orbits for subshifts of finite type: large intersections and emergence
• Dirac physical measures on saddle-type fixed points
• Irregular sets for piecewise monotonic maps
• Ergodic Average Of Typical Orbits And Typical Functions
• Historic and physical wandering domains for wild blender-horseshoes
• Lyapunov exponents for random maps
• Dissecting a resonance wedge on heteroclinic bifurcations
• Existence and non-existence of length averages for foliations
• Berger domains and Kolmogorov typicality of infinitely many invariant circles
• Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic
• Ultralimits of Birkhoff averages
• Distributional chaos in multifractal analysis, recurrence and transitivity
• Historic Behaviour for Random Expanding Maps on the Circle
• Takens' Last Problem and strong pluripotency
• Hénon maps: a list of open problems
• Entropy of irregular points that are not uniformly hyperbolic
• SRB measures for C $\infty$ surface diffeomorphisms
• Strange attractors and non wandering domains near a homoclinic cycle to a bifocus
• Physical-like Measures Coincide with Invariant Measures Supported on Chain Recurrent Classes