Henon Maps: Open Problems — Problem 33

Definitions and mathematical statement

Let $f\colon X\to X$ denote the underlying Hénon dynamical system. The supplied record does not specify the map, its parameters, the real or complex setting, or the invariant phase space $X$. The following describes the intended assertion using the usual variational meaning of equilibrium state; these missing choices must be supplied before it becomes a fully specified mathematical problem.

For a specified measurable dynamical system, let $\mathcal M_f(X)$ be the set of $f$-invariant Borel probability measures on $X$, meaning that $\mu(f^{-1}(A))=\mu(A)$ for every Borel set $A$. For a real-valued potential $\phi\colon X\to{\mathbb{R}}$, when the integrals and entropy expressions are well defined, put \[ P_f(\phi)=\sup_{\nu\in\mathcal M_f(X)}\left(h_\nu(f)+\int_X\phi\,{\,\mathrm{d}}\nu\right), \] where $h_\nu(f)$ is the measure-theoretic entropy. An equilibrium state for $\phi$ is a measure $\mu_\phi\in\mathcal M_f(X)$ attaining this supremum. The appropriate integrability conditions and any necessary finiteness conditions are not specified in the supplied record.

The requested assertion is that, for every sufficiently regular potential $\phi$ in the intended class, there exists exactly one equilibrium state $\mu_\phi$. The source suggests taking every Hölder continuous potential, possibly subject to a bound on \[ \max_X\phi-\min_X\phi. \] For a specified metric $d$ on $X$, Hölder continuity means that there exist $C\geq0$ and $0<\alpha\leq1$ such that \[ |\phi(x)-\phi(y)|\leq C\,d(x,y)^\alpha\qquad(x,y\in X). \] The metric, the admissible regularity class, whether an oscillation bound is required, its value and parameter dependence, and hypotheses ensuring that the displayed extrema exist are all unresolved in this record. Consequently, the unrestricted Hölder version and a version with a bounded oscillation must remain distinct possible targets.

Short English statement

For the intended Hénon system, prove that every sufficiently regular weight has a unique invariant probability measure maximizing entropy plus the integral of the weight. The source suggests Hölder weights and allows a possible restriction on their oscillation. The supplied excerpt omits the dynamical setting and the precise potential class, so the statement requires clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $801081$, source problem code catalogue entry, title: Henon Maps: Open Problems --- Problem 33. The registry attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open; that status is supplied metadata, not an independently verified mathematical assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Problem 33. Supplied source URL: https://arxiv.org/abs/2312.03907. The background records access on 2026-10-16 and extraction by arxiv_latex; these provenance claims are supplied and have not been independently verified here.