Multiplier-group subgroup conjecture for algebraic quasiperiodic flows

Let $T^n$ be the $n$ -torus, let $\phi$ be a quasiperiodic flow on $T^n$ generated by a constant vector field $X$ , and let $F$ be a real algebraic number field of degree $n$ over $\mathbb{Q}$ . Suppose that there is a nonzero real number $\vartheta$ such that the components of $\vartheta X$ form a basis of $F$ over $\mathbb{Q}$ . The multiplier-group subgroup conjecture. The multiplier group of $\phi$ is a subgroup of the group of units of the ring of integers in $F$ . This conjecture concerns the arithmetic structure of the smooth-conjugacy invariant associated with a quasiperiodic flow; the paper later says that the conjecture is reformulated and proved, so its resolution should be checked against the cited theorem.

Lennard F. Bakker formulated the multiplier-group subgroup conjecture for algebraic quasiperiodic flows. His 30 July 2003 paper says the conjectures are reformulated and proved in Theorems $3.4$ and $3.8$ , which directly concerns the stated one-way inclusion.

Known results

Bakker, 2004: a semiconjugacy theorem gives a finite-index subgroup relation for multiplier groups, partially supporting a broader converse conjecture.
Bakker, 2005: the broader characterization is described as partially validated for $n \ge 2$ and fully validated for $n=2$ ; this is distinct from the stated subgroup assertion.

30 July 2003 claimed proof

Bakker’s paper explicitly claims to reformulate and prove the relevant conjectures, with Theorems $3.4$ and $3.8$ supplying the stated result. The scan found no subsequent error report, retraction, referee assessment, or independent verification.

Current status (as of September 2026): the conjecture has a direct claimed proof in Bakker’s 2003 paper, but that proof is not independently verified in the supplied evidence; the broader converse characterization remains only partially established.

Multiplier-group subgroup conjecture for algebraic quasiperiodic flows

Primary source

Progress summary

Solutions 0