Local Typicality of the Newhouse Phenomenon

Definitions and mathematical statement

For a discrete-time system given by a $C^1$ map $f\colon M\to M$ on a finite-dimensional smooth manifold, a periodic orbit of least period $p\geq 1$ is a sink if every eigenvalue of $D(f^p)_x$, for a point $x$ on the orbit, has modulus strictly less than $1$. In this setting, the coexistence property discussed in the supplied literature assessment is \[ \mathcal N=\{f: f\text{ has infinitely many distinct periodic sink orbits}\}. \] This definition explains the discrete-time sink formulation; the source does not specify that this is the full intended class of systems.

Source question. Is the Newhouse phenomenon locally typical in the space of smooth dynamical systems?

A fully quantified mathematical formulation remains unresolved in the supplied record. It does not specify the underlying manifold or its dimension and compactness assumptions, whether the systems are maps, diffeomorphisms, or flows, the regularity class and topology, the meaning of "typical," or the neighborhoods and reference systems involved in "locally." In particular, residuality, germ-typicality, and Arnold prevalence must not be identified without an explicit definition. The record therefore does not determine a unique proposition about $\mathcal N$.

The supplied dated assessment describes coexistence of infinitely many sinks as $C^r$ germ-typical near dissipative bicycles for finite $r$, while describing stronger local typicality notions, including Arnold prevalence, and the smooth or analytic cases as remaining open. This is supplied curation, not an independently verified theorem or current status assessment. The allowed range of finite $r$, the meaning and hypotheses of a dissipative bicycle, and the parameter-family quantifiers for germ-typicality are not supplied. Here "smooth" also requires clarification, particularly whether it means $C^\infty$.

Short English statement

The question asks whether coexistence of infinitely many attracting periodic orbits is locally typical among smooth dynamical systems. Its precise meaning depends on an unspecified class of systems, topology, and notion of typicality. The supplied assessment reports partial progress for finite differentiability near dissipative bicycles, but its hypotheses and claimed remaining cases require source verification.

Sources

• Local UnsolvedMath registry, supplied complete record, permanent ID $30004744$, title Local Typicality of the Newhouse Phenomenon, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The registry gives proposed year $2021$ and names no individual proposer.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, pp. 1735--1803, DOI 10.4171/owr/2021/33. https://doi.org/10.4171/owr/2021/33. The background attributes the collection to workshop reports and their open-problem discussions.
• Supplied literature source labelled "germ-typicality theorem": https://doi.org/10.1016/j.aim.2022.108528.
• Supplied literature source labelled "primary preprint": https://arxiv.org/abs/2103.16697.
• Literature assessment checked at 2026-08-22: the supplied status is partially_solved. The assessment reports finite-$r$ $C^r$ germ-typicality near dissipative bicycles and describes stronger local typicality notions, including Arnold prevalence, and smooth or analytic cases as open. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.