Henon Maps: Open Problems — Problem 35

Definitions and mathematical statement

The supplied record does not define the underlying dynamical system, the potential $\phi$, or the measure $\mu_\phi$. Consequently, the following expresses the requested conclusion, with these objects left subject to their missing source hypotheses.

Write $f\colon X\to X$ for the intended underlying map. For a real-valued observable $g$ for which the integral exists, set \[ S_n(g)(x)=\sum_{j=0}^{n-1}g(f^j(x)),\qquad \langle\mu_\phi,g\rangle=\int_X g\,\mathrm d\mu_\phi, \qquad Z_n(g)=\frac{S_n(g)-n\langle\mu_\phi,g\rangle}{\sqrt n}. \] These formulas interpret the notation in the record; the map, its domain, the probability measure, and the integrability hypotheses must be recovered from the source context. Relative to the intended metric $d$ on $X$, a real-valued observable is Hölder continuous if there are constants $C\geq0$ and $\alpha>0$ such that \[ |g(x)-g(y)|\leq C\,d(x,y)^\alpha\qquad(x,y\in X). \] The intended metric, observable domain, and any boundedness or local versus global Hölder convention are not supplied.

Requested assertion, with unresolved source qualifications. For every intended Hölder continuous observable $g$, there exists $\sigma=\sigma(g)\geq0$ such that, if $\sigma^2>0$, then for every interval $I\subseteq{\mathbb{R}}$, \[ \lim_{n\to\infty}\mu_\phi\left(\{x\in X:Z_n(g)(x)\in I\}\right) =\frac{1}{\sqrt{2\pi\sigma^2}}\int_I \exp\!\left(-\frac{t^2}{2\sigma^2}\right)\,\mathrm dt. \] If $\sigma^2=0$, the displayed source conclusion specifies only \[ \lim_{n\to\infty}\mu_\phi\left(\{x\in X:Z_n(g)(x)\in(-\delta,\delta)\}\right)=1 \qquad(\delta>0). \] The positivity condition on $\delta$ makes explicit the neighborhood interpretation of this clause. Although the source introduces its formula with a quantifier over every interval, its zero-variance branch gives no conclusion for other intervals. A standard central limit theorem would express convergence in distribution to $\mathcal N(0,\sigma^2)$, interpreting $\mathcal N(0,0)$ as the point mass at zero; this interpretation does not imply convergence of probabilities for every interval with zero on its boundary. The intended zero-variance formulation therefore requires clarification.

Short English statement

The problem asks whether the intended measure $\mu_\phi$ satisfies a central limit theorem for every real-valued Hölder observable, with variance allowed to vanish. The supplied excerpt omits the dynamical and measure hypotheses and leaves its assertion for arbitrary intervals incomplete in the zero-variance case.

Sources

• Local UnsolvedMath registry: permanent ID $801083$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Problem 35." The record attributes the problem to Berger and Bedford, with proposed year $2023$.
• Berger, Bedford, Henon Maps: A List of Open Problems (2023), Problem 35. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports extraction by arxiv_latex and access on 2026-10-16; these are supplied provenance claims, not independently verified access or source verification.
• The local record labels the problem open. This is a supplied status label, not an independently verified assessment of the literature.