Henon Maps: Open Problems — Question 17

Definitions and mathematical statement

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, equipped with normalized Lebesgue measure $m$, and let $\mathbb{T}_{\mathrm{irr}}=\{x+\mathbb{Z}:x\in\mathbb{R}\setminus\mathbb{Q}\}$. The supplied question begins with the qualification "If the answer is yes," referring to an antecedent question absent from the record. Subject to that unresolved qualification, retain the source's notation \[ \Theta=\{\theta\in\mathbb{T}_{\mathrm{irr}}:H_2\text{ has a Herman ring with rotation number }\theta\}. \] The supplied record does not define $H_2$, specify whether it denotes a map or a class of maps, or define the intended notion of a Herman ring. Thus the membership condition above remains source-dependent and does not yet specify a fully determined set.

Question (Berger--Bedford, Question 17). With those source-dependent terms resolved and the preceding affirmative-answer qualification satisfied, does \[ m(\Theta)>0 \] hold? More specifically, is the class $0+\mathbb{Z}$ a Lebesgue density point of $\Theta$? Explicitly, writing \[ I_r=\{x+\mathbb{Z}:x\in(-r,r)\},\qquad 0<r<\tfrac12, \] the latter question asks whether \[ \lim_{r\downarrow0}\frac{m(\Theta\cap I_r)}{2r}=1. \] These measure formulations presuppose Lebesgue measurability of $\Theta$, which the supplied record does not establish. Density is taken in the ambient circle $\mathbb{T}$; it does not require $0+\mathbb{Z}$ to belong to $\Theta$. The density assertion is stronger than positive measure, rather than an equivalent reformulation.

Short English statement

Conditionally on an unspecified preceding question having an affirmative answer, Berger and Bedford ask whether the irrational rotation numbers realized by Herman rings for the source's undefined $H_2$ form a set of positive Lebesgue measure, and more specifically whether that set has density one at zero. The missing antecedent and definition of $H_2$ require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801065$, exact source problem code catalogue entry, titled "Henon Maps: Open Problems --- Question 17." The registry 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 or literature assessment.
• Berger, Bedford, Hénon Maps: A List of Open Problems (2023), Question 17. Supplied source URL: https://arxiv.org/abs/2312.03907. The provenance records extraction by arxiv_latex and access on 2026-10-16. This access date and attribution are supplied provenance and have not been independently verified.