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Shin Kiriki

Publications and source records attributed to Shin Kiriki.

At least 19 recordsLinked to original sources
arXiv2026

A robust obstruction to full strong pluripotency for wild blender-horseshoes

Suppose that $M$ is a closed manifold of dimension greater than two and $r\geq 2$. We show that there exists a $C^r$-diffeomorphism $f:M\longrightarrow M$ with a wild affine blender-horseshoe $\Lambda_f$ which is $C^r$-robustly and strongly pluripotent for $\Lambda_f^{(\mathrm{mj})}$ but not for $\Lambda_f$, where $\Lambda_f^{(\mathrm{mj})}$ is the subset of $\Lambda_f$ consisting of elements with the majority condition. Thus, within the present affine blender-horseshoe family in [KNS], there is a robust obstruction to extending the strong pluripotency from $\Lambda_f^{(\mathrm{mj})}$ to the whole horseshoe.

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arXiv2026

Non-existence of Lyapunov exponents in the Newhouse domain

We show that within the Newhouse domain of $C^r$ surface diffeomorphisms ($r \in [2,\infty )$), there exists a dense subset $\mathcal D$ such that for any $f \in \mathcal D$, Lyapunov exponents fail to exist for all points in some open set $U$ and all nonzero tangent vectors in some open cone $V \subset \mathbb{R}^2$. This demonstrates that the non-existence of Lyapunov exponents is a persistent phenomenon in the setting of robust homoclinic tangencies. The proof relies on constructing diffeomorphisms exhibiting specific oscillatory return times near a homoclinic tangency, incorporating techniques from Newhouse theory and recent results on Lyapunov irregularity, alongside several refinements and new arguments.

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arXiv2024

Takens' Last Problem and strong pluripotency

We consider the concept of strong pluripotency of dynamical systems for a hyperbolic invariant set, as introduced in [KNS]. To the best of our knowledge, for the whole hyperbolic invariant set, the existence of robust strongly pluripotent dynamical systems has not been proven in previous studies. In fact, there is an example of strongly pluripotent dynamical systems in [CV01], but its robustness has not been proven. On the other hand, robust strongly pluripotent dynamical systems for some proper subsets of hyperbolic sets had been found in [KS17, KNS]. In this paper, we provide a combinatorial way to recognize strongly pluripotent diffeomorphisms in a Newhouse domain and prove that they are $C^r$-robust, $2\leq r< \infty$. More precisely, we prove that there is a 2-dimensional diffeomorphism with a wild Smale horseshoe which has a $C^r$ neighborhood $\mathcal{U}_0$ where all elements are strongly pluripotent for the whole Smale horseshoe. Moreover, it follows from the result that any property, such as having a non-trivial physical measure supported by the Smale horseshoe or having historic behavior, is $C^r$-persistent relative to a dense subset of $\mathcal{U}_0$.

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arXiv2024

Pluripotency of wandering dynamics

This paper isolates a perturbative mechanism, which we call \emph{pluripotency}, by which the symbolic and statistical behavior of prescribed orbits in a uniformly hyperbolic set can be realized, after an arbitrarily small perturbation, along the forward orbits of all points in a set of positive Lebesgue measure. In this sense, pluripotency provides a way of reprogramming dynamics from both statistical and geometric viewpoints: the empirical measures of all points in a positive-measure set can be made to asymptotically follow those of a prescribed orbit in the hyperbolic set. We first give an abstract criterion, formulated in terms of symbolic itinerary descriptions, which is equivalent to a strong form of pluripotency. We then prove that this mechanism occurs robustly in higher-dimensional non-hyperbolic dynamics. More precisely, for every $2\le r<\infty$ and $\dim M\ge 3$, there exists a $C^r$-open set of diffeomorphisms with wild blender-horseshoes such that every diffeomorphism in this open set is strongly pluripotent for a dense invariant subset of the blender-horseshoe. As applications, this yields dense classes of diffeomorphisms with non-trivial Dirac physical measures and with historic wandering domains inside the same open set, providing a new mechanism related to Takens' last problem.

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arXiv2022

Historic and physical wandering domains for wild blender-horseshoes

We present diffeomorphisms of wild blender-horseshoes which belong to $C^r$ $(1\leq r<\infty)$ closures of two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure supported by saddle periodic orbit. It is a non-trivial extension of Colli-Vargas' model [CV01] to the higher dimensional dynamics with the use of wild blender-horseshoes.

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arXiv2022

Abundance of observable Lyapunov irregular sets

Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke [OY2008], whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas [CV2001], as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. We can construct such positive Lebesgue measure sets both as the time averages exist and do not exist on it.

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arXiv2022

Emergence via non-existence of averages

Inspired by a recent work by Berger, we introduce the concept of pointwise emergence. This concept provides with a new quantitative perspective into the study of non-existence of averages for dynamical systems. We show that high pointwise emergence on a large set appears for abundant dynamical systems: Any continuous maps on a compact metric space with the specification property have super-polynomial pointwise emergence on a residual subset of the state space. Furthermore, there is a dense subset of any Newhouse open set each element of which has super-polynomial pointwise emergence on a positive Lebesgue measure subset of the state space.

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arXiv2019

Historic behavior in non-hyperbolic homoclinic classes

We show that $C^1$-generically for diffeomorphisms of manifolds of dimension $d\geq3$, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behavior.

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arXiv2018

Historic behaviour for nonautonomous contraction mappings

We consider a parametrised perturbation of a $\mathscr C^r$ diffeomorphism on a closed smooth Riemannian manifold with $r\geq 1$, modeled by nonautonomous dynamical systems. A point without time averages for a (nonautonomous) dynamical system is said to have historic behaviour. It is known that for any $\mathscr C^r$ diffeomorphism, the observability of historic behaviour, in the sense of the existence of a positive Lebesgue measure set consisting of points with historic behaviour, disappears under absolutely continuous, independent and identically distributed (i.i.d.) noise. On contrast, we show that the observability of historic behaviour can appear by a non-i.i.d. noise: we consider a contraction mapping for which the set of points with historic behaviour is of zero Lebesgue measure and provide an absolutely continuous, non-i.i.d. noise under which the set of points with historic behaviour is of positive Lebesgue measure.

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arXiv2018

Moduli of 3-dimensional diffeomorphisms with saddle-foci

We consider a space $\mathcal{U}$ of 3-dimensional diffeomorphisms $f$ with hyperbolic fixed points $p$ the stable and unstable manifolds of which have quadratic tangencies and satisfying some open conditions and such that $Df(p)$ has non-real expanding eigenvalues and a real contracting eigenvalue. The aim of this paper is to study moduli of diffeomorphisms in $\mathcal{U}$. We show that, for a generic element $f$ of $\mathcal{U}$, all the eigenvalues of $Df(p)$ are moduli and the restriction of a conjugacy homeomorphism to a local unstable manifold is a uniquely determined linear conformal map.

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arXiv2017

Non-trivial wandering domains for heterodimensional cycles

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.

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arXiv2016

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

In this paper, we give an answer to a $C^{r}$ $(2\leq r <\infty)$ 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.

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arXiv2013

Blenders in center unstable Hénon-like families: with an application to heterodimensional bifurcations

We give an explicit family of polynomial maps called center unstable Hénon-like maps and prove that they exhibits blenders for some parametervalues. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.

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arXiv2012

$C^2$-robust heterodimensional tangencies

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 $\dim(M)\geq 3$ each element of which has a non-degenerate heterodimensional tangency on a $C^2$ robust heterodimensional cycle.

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arXiv2012

Existence of generic cubic homoclinic tangencies for Hénon maps

In this paper, we show that the Hénon map $φ_{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko-Shilnikov-Turaev [12]-[16]. Combining this fact with theorems in Kiriki-Soma [20], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

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arXiv2011

Stabilization of heterodimensional cycles

We consider diffeomorphisms $f$ with heteroclinic cycles associated to saddles $P$ and $Q$ of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $P$ and $Q$. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized.

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arXiv2010

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

Let $\{f_{a,b}\}$ be the (original) Hénon family. In this paper, we show that, for any $b$ near $0$, there exists a closed interval $J_b$ which contains a dense subset $J'$ such that, for any $a\in J'$, $f_{a,b}$ has a quadratic homoclinic tangency associated with a saddle fixed point of $f_{a,b}$ which unfolds generically with respect to the one-parameter family $\{f_{a,b}\}_{a\in J_b}$. By applying this result, we prove that $J_b$ contains a residual subset $A_b^{(2)}$ such that, for any $a\in A_b^{(2)}$, $f_{a,b}$ admits the Newhouse phenomenon. Moreover, the interval $J_b$ contains a dense subset $\tilde A_b$ such that, for any $a\in \tilde A_b$, $f_{a,b}$ has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously. Dedicated to the memory of Floris Takens (Nov. 12, 1940 - Jun. 20, 2010).

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