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Masayuki Asaoka

Publications and source records attributed to Masayuki Asaoka.

At least 19 recordsLinked to original sources
arXiv2026

Positive hyperbolic periodic orbits for area-preserving maps and Reeb flows

We prove that a strongly nondegenerate Reeb vector field on a closed contact three-manifold has infinitely many simple positive hyperbolic periodic orbits if it has at least three simple periodic orbits. The main ingredient is a result for area-preserving maps on compact surfaces, possibly with boundary. We prove that, under a natural boundary index condition, an area-preserving nondegenerate map with infinitely many periodic points has infinitely many positive hyperbolic periodic points in the interior. The proof combines this surface result with the broken book decomposition theorem of Colin--Dehornoy--Rechtman.

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arXiv2024

Length averages for codimension one foliations

In this paper we study geometrical and dynamical properties of codimension one foliations, by exploring a relation between length averages and ball averages of certain group actions. We introduce a new mechanism, which relies on the group structure itself, to obtain irregular behavior of ball averages for certain non-amenable group actions. Several geometric realization results show that any such groups can appear connected with the topology of leaves which are connected sums of plugs with a special geometry, namely nearly equidistant boundary components. This is used to produce the first examples of codimension one $\mathcal C^\infty$ regular foliations on a compact Riemannian manifold $M$ for which the length average of some continuous function does not exist on a non-empty open subset of $M$.

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arXiv2024

Oriented Birkhoff sections of Anosov flows

This paper gives 3 different proofs (independently obtained by the 3 authors) of the following fact: given an Anosov flow on an oriented 3 manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is $\mathbb{R}$-covered positively twisted.

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arXiv2023

Area-preserving diffeomorphisms on the disk and positive hyperbolic orbits

In this paper, we prove that if an area-preserving non-degenerate diffeomorphism on the open disk which extend smoothly to the boundary with non-degeneracy has at least 2 interior periodic points, then there are infinitely many positive hyperbolic periodic points on the interior. As an application, we prove that if a non-degenerate universally tight contact 3-dimentional lens space has a Birkhoff section of disk type and at least 3 simple periodic orbits, there are infinitely many simple positive hyperbolic orbits. In particular, we have that a non-degenerate dynamically convex contact 3-sphere has either infinitely many simple positive hyperbolic orbits or exactly two simple elliptic orbits, which gives a refinement of the result proved by Hofer, Wysocki and Zehnder in \cite{HWZ2} under non-degeneracy.

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arXiv2022

Goodman-Fried surgery, Birkhoff section, and R-covered Anosov flows

In this paper, we show that any topologically transitive Anosov flow on a closed three-dimensional manifold with the orientable weak stable and unstable foliations turns into an R-covered one by a Goodman-Fried surgery along periodic points. When a Birkhoff section of the Anosov flow is given, we can choose the periodic points from the boundary of the Birkhoff section.

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arXiv2021

Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any higher dimensional manifold, using blenders. The former implies that an analog of Moreira's theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and A.Raibekas.

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arXiv2018

Fast growth of the number of periodic points arising from heterodimensional connections

We consider C^r-diffeomorphisms of a compact smooth manifold having a pair of robust heterodimensional cycles where r is a positive integer or infinity. We prove that if certain conditions about the signatures of non-linearities and Schwarzian derivatives of the transition maps are satisfied, then by giving C^r arbitrarily small perturbation, we can produce a periodic point at which the first return map in the center direction is C^r-flat. As a consequence, we will prove that C^r-generic diffeomorphisms in the neighborhood of the initial diffeomorphism exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions on non-linearities and the Schwarzian derivatives.

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arXiv2017

Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.

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arXiv2016

A $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces

We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.

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arXiv2015

Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

We consider semigroup actions on the unit interval generated by strictly increasing $C^r$-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on the non-linearity and the Schwarzian derivative, the above semigroups exhibit, $C^r$-generically for any r, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a $C^r$-generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.

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arXiv2013

Deformation of locally free actions and the leafwise cohomology

This is a note of the author's lectures at "Advanced courses in Foliation" in the research program "Foliation", which was held at the Centre de Recerca Mathematica in the May of 2010. In this note, we discuss about the relationship between deformation of actions of Lie groups and the leafwise cohomology of the orbit foliation.

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arXiv2012

Rigidity of certain solvable actions on the sphere

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure.

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arXiv2011

Non-homogeneous locally free actions of the affine group

We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a non-homogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.

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arXiv2009

Regular projectively Anosov flows on three-dimensional manifolds

We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of $T^2 \times I$-models. We also apply our method to rigidity problems of some group actions.

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