Common Basin Boundaries in a Transfer-Operator System

Definitions and mathematical statement

The supplied record concerns a particular transfer-operator dynamical system in the parameter regime $6<B\leq16$, with fixed points denoted by $u_{-r_*},u_0,u_{r_*}$. It does not specify the evolution operator, its state space, the topology, the definition of $r_*$, or the fixed-point functions. Consequently, the following expresses the intended conjecture conditionally on those missing source-specific data; it does not define the underlying system.

For each admissible $B$, write $T_B:X_B\to X_B$ for the unspecified evolution on its unspecified topological state space. Suppress any dependence of the fixed points and $r_*$ on $B$. The stated fixed-point conditions are \[ T_Bu_{-r_*}=u_{-r_*},\qquad T_Bu_0=u_0,\qquad T_Bu_{r_*}=u_{r_*}. \] Under the interpretation that a basin consists of states whose forward iterates converge to the indicated fixed point, define \[ \mathcal B_B(u)=\left\{x\in X_B:T_B^{\,n}(x)\longrightarrow u\text{ as }n\to\infty\right\}, \] where $T_B^{\,0}$ is the identity and convergence must be understood in the topology intended by the source. For $A\subseteq X_B$, its boundary relative to $X_B$ is \[ \partial_{X_B}A=\overline{A}^{\,X_B}\setminus\operatorname{int}_{X_B}(A). \] Interpreting "the common boundary" as equality with both complete basin boundaries, the intended assertion is \[ \text{for every }B\in(6,16],\qquad \mathcal B_B(u_0) =\partial_{X_B}\mathcal B_B(u_{-r_*}) =\partial_{X_B}\mathcal B_B(u_{r_*}). \] The endpoint $B=6$ is excluded and $B=16$ is included. The record does not resolve whether "common boundary" instead means only the intersection of the two boundaries. The definitions of the system, admissible states, convergence, and ambient boundary topology must be recovered from the cited report before this can serve as a complete mathematical formulation.

Short English statement

For the specified transfer-operator system and every $6<B\leq16$, the question asks whether the basin of $u_0$ equals the boundary of each of the basins of $u_{-r_*}$ and $u_{r_*}$. The system, topology, and meaning of $r_*$ are missing from the supplied record, so this is a conditional formulation requiring source clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30001370$, source problem code catalogue entry, title "Common Basin Boundaries in a Transfer-Operator System." The supplied complete record is the local evidence for this draft.
• Supplied source citation: Mini-Workshop: Spectrum of Transfer Operators -- Recent Developments and Applications (2010), Oberwolfach Reports, pp. 2699--2730, DOI 10.4171/owr/2009/49. Supplied source URL: https://doi.org/10.4171/owr/2009/49. The record describes Oberwolfach Reports as reports of mathematical workshops and their open-problem discussions. No individual proposer is supplied; the recorded proposed year is 2009.
• Supplied literature source, labeled "original Oberwolfach report 4132": https://doi.org/10.4171/owr/2009/49.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes it as an explicitly stated basin-boundary conjecture in a specific dynamical system, and states that the extracted report supplies no proof and that the available literature does not establish the asserted common-boundary description. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, accompanied by a claim that the statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and open target. That verification claim is part of the supplied record; the report has not been independently consulted for this draft.