Kirby Problem 5.2

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a homeomorphism $h$ of $\mathbb R^n$, the orbit of $x$ is $\mathcal O_h(x)=\{h^m(x):m\in\mathbb Z\}$, using positive and negative iterates. Compact-open convergence of maps $h^{m_j}$ to the identity means uniform convergence to the identity on every compact subset of $\mathbb R^n$. The closed ball is $B^n=\{x\in\mathbb R^n:\|x\|\leq1\}$.

Statement recorded in the supplied source.

For every $n\geq1$, if a homeomorphism (or smooth diffeomorphism) $h:\mathbb R^n\to\mathbb R^n$ satisfies \[ \sup_{x\in\mathbb R^n}\operatorname{diam}\mathcal O_h(x)\leq1, \qquad h^{m_j}\longrightarrow\operatorname{id}_{\mathbb R^n} \text{ in the compact-open topology for some }m_j\to\infty, \] must $h$ be the identity? Also resolve the ball version: if $h:B^n\to B^n$ is a homeomorphism equal to the identity on $\partial B^n$ and some positive powers $h^{m_j}$ with $m_j\to\infty$ converge to the identity, must $h=\operatorname{id}_{B^n}$? [S2]

Short English statement

Must a recurrent homeomorphism with uniformly bounded orbits be trivial, and must a recurrent homeomorphism of a ball fixing its boundary be trivial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3009 (KP-5.2). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 5.2, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.