Henon Maps: Open Problems — Problem 36

Definitions and mathematical statement

The supplied record does not define the phase space $X$, its metric, the underlying dynamical map, the potential $\phi$, or the measure $\mu_\phi$. Consequently, the following is a conditional transcription of the requested statement; these missing data and their hypotheses must be recovered before it becomes a fully specified problem.

Write $T:X\to X$ for the underlying map implicit in the source, and use the Birkhoff-sum convention \[ S_n(g)(x):=\sum_{j=0}^{n-1}g(T^j(x)),\qquad n\geq 1. \] For a specified metric $d$ on $X$, a real-valued observable $g:X\to{\mathbb{R}}$ is Hölder continuous if there exist $\alpha\in(0,1]$ and $C\geq0$ such that \[ |g(x)-g(y)|\leq C d(x,y)^\alpha\qquad(x,y\in X). \] The source does not specify the metric, observable domain, or intended Hölder convention. Write $\langle\mu_\phi,g\rangle=\int_X g\,\mathrm d\mu_\phi$, whenever this integral is defined. A coboundary ordinarily means an observable expressible as $g=u\circ T-u$; the source does not specify the permitted regularity or integrability of $u$, or whether the equality is pointwise or almost everywhere. Its non-coboundary qualification therefore remains unresolved.

With the intended setting and these conventions supplied, the requested assertion is: for every real-valued Hölder continuous observable $g$ satisfying \[ \int_X g\,\mathrm d\mu_\phi=0 \] and which is not a coboundary in the intended sense, there exist an open interval $I\subseteq{\mathbb{R}}$ containing $0$, a strictly convex function $c:I\to[0,\infty)$, and $\delta>0$ with $(0,\delta)\subseteq I$, such that \[ c(t)=0\quad\Longleftrightarrow\quad t=0\qquad(t\in I) \] and, for every $\epsilon\in(0,\delta)$, \[ \lim_{n\to\infty}\frac1n\log\mu_\phi\!\left( \left\{x\in X:\frac{S_n(g)(x)}n>\epsilon\right\} \right)=-c(\epsilon). \] Here $\log0=-\infty$. The function $c$, interval $I$, and threshold $\delta$ may depend on $g$ and the intended dynamical data. The source calls this the Large Deviation Principle; its displayed target specifies positive-tail asymptotics for sufficiently small positive thresholds and does not explicitly state a full large deviation principle for arbitrary measurable sets.

Short English statement

For each centered Hölder observable that is not a coboundary, prove that sufficiently small positive deviations of its time average have exponential probabilities governed by a nonnegative strictly convex function vanishing exactly at zero. The dynamical setting, measure, potential, and coboundary convention are missing from the supplied record and require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $801084$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Problem 36." The supplied record attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Problem 36. Supplied source URL: https://arxiv.org/abs/2312.03907. The background records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source has not been independently consulted for this draft.