arXiv subjects

Shuntaro Tomizawa

Publications and source records attributed to Shuntaro Tomizawa.

3 recordsLinked to original sources

arXiv2026

$C^1$-robust strong pluripotency for blender-horseshoes

Suppose that $M$ is a closed manifold of dimension greater than two. We show that there exists a $C^1$-diffeomorphism $f_0:M\longrightarrow M$ with a wild affine blender-horseshoe $Λ_{f_0}$ such that any element $f$ of $\mathrm{Diff}^1(M)$ sufficiently $C^1$-close to $f_0$ is strongly pluripotent for the continuation $Λ_f^{(\mathrm{mj})}$ of $Λ_{f_0}^{(\mathrm{mj})}$, where $Λ_{f_0}^{(\mathrm{mj})}$ is the dense subset of $Λ_{f_0}$ consisting of elements with majority condition.

math.DS↗

arXiv2026

Infinitely Many Attracting Periodic Circles in Higher Dimensions

We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable and unstable eigenvalues closest to $1$ in modulus are real and simple. One heteroclinic connection is transverse and the other is non-transverse, and the product of those two eigenvalues is less than $1$ at one point and greater than $1$ at the other. Arbitrarily close to such a map, there are open sets in which a residual subset of diffeomorphisms has infinitely many attracting normally hyperbolic periodic circles. The proof uses a rescaling to the standard Hénon map and a corrected formula for the Lyapunov coefficient on its Neimark-Sacker (Andronov-Hopf) line.

math.DS↗

arXiv2025

Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations

We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|λγ| = 1$. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus $|λγ|$, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map $f$ can be $C^{r}$-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case.

math.DS↗