Henon Maps: Open Problems — Question 15

Definitions and mathematical statement

For a map $f\colon X\to X$, a periodic cycle of least period $n\geq 1$ is an orbit $C=\{x,f(x),\ldots,f^{n-1}(x)\}$ such that $f^n(x)=x$ and $f^k(x)\neq x$ for every integer $1\leq k<n$.

The supplied source, Question 15 of Berger and Bedford's Henon Maps: A List of Open Problems (2023), asks:

Does $H_2$ have both small saddle cycles and small elliptic cycles?

The supplied record does not define $H_2$, its phase space, or the terms "small," "saddle," and "elliptic" in this question. In particular, it does not specify whether smallness concerns orbit diameter, location near a distinguished point, or another quantity, nor whether the question requires arbitrarily small cycles or cycles satisfying a fixed bound. The domain and multiplier conditions governing the saddle and elliptic classifications are also absent. Consequently, the displayed question preserves the supplied formulation, but a fully specified mathematical statement and its quantifiers require the missing source context.

Short English statement

Berger and Bedford ask whether the map denoted by $H_2$ has both small saddle periodic cycles and small elliptic periodic cycles. The supplied excerpt leaves the map and the relevant meaning and quantifiers of smallness unspecified.

Sources

• Local UnsolvedMath source registry: permanent ID $801063$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 15." The record attributes the question to Berger and Bedford, with proposed year 2023, and labels its status as open; this is a supplied registry label, not an independently verified mathematical or literature assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 15. 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, not a claim of independent source access or verification.