Henon Maps: Open Problems — Question 37

Definitions and mathematical statement

Let $J$ denote the Julia set referred to in the source question, and let $f_*$ denote the source's operator on functions on $J$. The supplied record does not specify the map $f$, the definition of $J$, or the action and normalization of $f_*$. For a potential $\phi$, write formally \[ \mathcal L_\phi g=f_*\left(e^{\phi-P(\phi)}g\right), \] whenever these operations are defined. The admissible potentials and the quantity $P(\phi)$ are not defined in the supplied record. An invariant line for an operator $T$ means a one-dimensional linear subspace $E$ satisfying $T(E)\subseteq E$.

Question (Berger and Bedford, Question 37, 2023). Do there exist norms for functions on $J$ which are bounded on Hölder continuous functions, contract on the complement of an invariant line under $f_*$, or, more generally, under $\mathcal L_\phi$, and for which the contraction is stable under perturbation of $\phi$?

The question retains unresolved qualifications from the supplied source. The function space, scalar field, metric on $J$, Hölder exponent, meaning of boundedness, invariant line, and meaning of its complement are unspecified. For example, a continuous-inclusion interpretation of boundedness would require an estimate $\|g\|\leq C\|g\|_{C^\alpha(J)}$ for specified $\alpha$, metric, and constant $C$; this interpretation is not fixed by the record. Likewise, contraction could refer to an operator-norm bound strictly below one on a specified invariant complement, decay of iterates, or contraction of an induced quotient operator. The record does not select among these meanings. The topology of perturbations of $\phi$, the permitted dependence of the norm and invariant line on $\phi$, and any uniformity of contraction constants also remain unspecified. Consequently, this is a statement-only transcription of the existence question, requiring these choices before a fully quantified formulation can be given.

Short English statement

Berger and Bedford ask whether suitable norms on functions on the Julia set make the transfer operator, or its displayed potential-weighted version, contract away from an invariant line, with contraction persisting under perturbations of the potential. The supplied record leaves essential operator, function-space, and stability specifications unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $801085$, exact source problem code catalogue entry, titled Henon Maps: Open Problems --- Question 37. The record attributes the question to Berger and Bedford, with proposed year 2023, and labels it open; this is a supplied status label, not an independently verified mathematical or literature assessment.
• Berger and Bedford, Hénon Maps: A List of Open Problems (2023), Question 37. Supplied source URL: https://arxiv.org/abs/2312.03907. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access or verification.