Several Complex Variables Problem List — Problem 69

Definitions and mathematical statement

Unresolved context. The supplied statement does not define the measure $\mu$, the underlying space, or the dynamical system. Consequently, it does not determine a standalone mathematical conjecture.

The following notation gives only a conditional template. If $X$ is the intended topological space and $\mu$ is a nonnegative Borel measure on $X$, define \[ \operatorname{supp}_X(\mu)=\{x\in X:\mu(U)>0\text{ for every open neighbourhood }U\text{ of }x\}. \] If $f:X\to X$ is the intended dynamical map, a periodic point is a point $p$ satisfying $f^n(p)=p$ for some integer $n\geq 1$; its periodic orbit is $\{p,f(p),\ldots,f^{n-1}(p)\}$. Let $\mathcal R_f$ denote the union of the repelling periodic orbits, with the meaning of "repelling" still requiring the omitted dynamical context.

The supplied question is then represented schematically by \[ \operatorname{supp}_X(\mu)\stackrel{?}{=}\overline{\mathcal R_f}^{\,X}. \] The choices of $X$, $f$, and $\mu$, their hypotheses and quantifiers, the intended closure topology, and the definition of repelling are not supplied. In particular, the nonnegative Borel measure convention above is conditional notation, not a recovered hypothesis of the source problem.

Short English statement

The question asks whether the support of an unspecified measure $\mu$ equals the closure of the union of the repelling periodic orbits of an unspecified dynamical system. The omitted context is necessary for a precise statement. The registry labels this problem open; that label is supplied metadata, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $800270$, source code catalogue entry, title Several Complex Variables Problem List --- Problem 69.
• Supplied source attribution: Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 69. Supplied source URL: https://arxiv.org/abs/math/9509202.
• The record reports access on 2026-10-16 and extraction by arxiv_latex. These provenance details are supplied metadata; the source has not been independently consulted or verified for this draft.