Teruhiko Soma

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

Abstract In this paper, we give an answer to a C r ( 2 ≤ r ∞ ) version of the open problem of Takens in [42] 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 [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 Henon family.

Existence of ruled wrappings in hyperbolic 3–manifolds

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

C2-robust heterodimensional tangencies

In this paper, we give sufficient conditions for the existence of C 2 robust heterodimensional tangency, and present a non-empty open set in Diff 2 (M) withdimM ! 3eachelementofwhichhasanon-degenerateheterodimensional tangency on a C 2 robust heterodimensional cycle.

Existence of generic cubic homoclinic tangencies for Hénon maps

In this paper, we show that the Henon mapa;b has a generically unfolding cubic tangency for some .a; b/ arbitrarily close to . 2; 0/ by applying results of Gonchenko, Shilnikov and Turaev (On models with non-rough Poincare homoclinic curves. Physica D 62(1-4) (1993), 1-14; Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. Chaos 6(1) (1996), 15-31; On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle. Proc. Steklov Inst. Math. 216 (1997), 70-118; Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. 67 (1999), 69-128, translation in J. Math. Sci. 105 (2001), 1738-1778; Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20 (2007), 241-275). Combining this fact with theorems in Kiriki and Soma (Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies. Nonlinearity 21(5) (2008), 1105-1140), one can observe the new phenomena in the Henon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies

In this paper, we study a two-parameter family {μ,ν} of two-dimensional diffeomorphisms such that 0,0 = has a cubic homoclinic tangency unfolding generically which is associated with a dissipative saddle point. Our first theorem presents an open set in the μν-plane with such that for any there exists a one-parameter subfamily of {μ,ν} passing through and exhibiting cubically related persistent contact-making and contact-breaking quadratic tangencies. Moreover, the second theorem shows that any such two-parameter family satisfies Wang–Youngs conditions which guarantee that some μ,ν arbitrarily near exhibits a cubic polynomial-like strange attractor with an SRB measure.

Parameter-shifted shadowing property of Lozi maps

In this article, we study a certain shadowing property on well-known two-dimensional maps presented by Lozi with the y-axis as their singularity set and strange attractors. Our main theorem shows that there exists a nonempty open subset of the Misiurewicz domain such that, for any , the Lozi map L a ,u2009 b with strange attractor has the parameter-shifted shadowing property.

Coexistence of invariant sets with and without SRB measures in Henon family

Let {fa,b} be the (original) Henon family. In this paper, we show that, for any b near 0, there exists a closed interval Jb which contains a dense subset J such that, for any a J, fa,b has a quadratic homoclinic tangency associated with a saddle fixed point of fa,b which unfolds generically with respect to the one-parameter family . By applying this result, we prove that Jb contains a residual subset such that, for any , fa,b admits the Newhouse phenomenon. Moreover, the interval Jb contains a dense subset such that, for any , fa,b has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously.

Non-trivial wandering domains for heterodimensional cycles*

We present a sufficient condition for three-dimensional diffeomorphisms having heterodimensional cycles under which the diffeomorphisms can be arbitrarily approximated by diffeomorphisms with non-trivial contracting wandering domains. Moreover the union of

Heterodimensional tangencies on cycles leading to strange attractors

omega

Scott's rigidity theorem for seifert fibered spaces; Revisited

-limit set of all points in the domains is a nonhyperbolic transitive Cantor set without periodic points.