Abstract. In this paper, we give an answer to a C r (2 ≤ r < ∞) version of the open problem of Takens in [41] 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 ColliVargas [6,§2]. 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 [39] which is concerned with wandering domains for Hénon family.

No abstract

The possibility of the existence of a traversible wormhole solution which does not break the weak energy condition is examined in a higher-derivative theory of gravity. No such solution is found, suggesting that a Lorentzian wormhole without the violation of the weak energy condition is incompatible with a wide class of gravitational theories. We show this in two simple examples of spherically symmetric wormhole solutions. PACS numberb): 04.20.Me

We present a short elementary proof of an existence theorem of certain CAT(−1)surfaces in open hyperbolic 3-manifolds. The main construction lemma in Calegari and Gabai's proof of Marden's Tameness Conjecture can be replaced by an applicable version of our theorem. Finally, we will give a short proof of the conjecture along their ideas. 57M50; 30F40

No abstract

No abstract

We present a sufficient condition for three-dimensional diffeomorphisms having heterodimensional cycles to be approximated arbitrarily well by diffeomorphisms with non-trivial contracting wandering domains via several perturbations. The key idea is to show that diffeomorphisms with heterodimensional cycles associated with saddle points with non-real eigenvalues can be approximated by diffeomorphisms with generalized homoclinic tangencies presented by Tatjer. The generalized homoclinic tangency is an organizing center including a Bogdanov-Takens bifurcation, by which one can obtain non-trivial contracting wandering domains together with a Denjoy-like construction.

In this paper, we give sufficient conditions for the existence of C 2 robust heterodimensional tangency, and present a nonempty open set in Diff 2 (M ) with dim M ≥ 3 each element of which has a non-degenerate heterodimensional tangency on a C 2 robust heterodimensional cycle.