Complete topological ($C^0$) conjugacy classification of linear operators on finite-dimensional spaces

After classifying finite-dimensional linear flows up to topological equivalence, Berger and Wynne pointed to the discrete-time analogue as the remaining frontier: classifying linear operators $A:X\to X$ up to conjugacy by a homeomorphism $h$ with $h\circ A=B\circ h$, a problem they described as significantly harder than its continuous-time counterpart and 'to some extent still unresolved'. The problem is completely solved when the operator's spectrum contains no root of unity (Kuiper–Robbin 1973 for real spaces, later extended to complex spaces), where topological conjugacy is decided by finitely many elementary data. In the roots-of-unity regime, however, topological conjugacy genuinely diverges from linear similarity — so-called nonlinear similarity, first occurring in dimension six — and despite deep surgery-theoretic criteria of Cappell–Shaneson, Hsiang–Pardon, and Hambleton–Pedersen for finite cyclic groups, no complete effective classification of linear operators up to $C^0$ conjugacy is known. That precise open core remains intact as of 2026.

Background

For a discrete-time linear system $x_{k+1}=Ax_k$ on a finite-dimensional real or complex vector space, the coarsest natural notion of equivalence is topological conjugacy ($C^0$ conjugacy): linear maps $A:X\to X$ and $B:Y\to Y$ are topologically conjugate if there exists a homeomorphism $h:X\to Y$ with $h(Ax)=Bh(x)$ for all $x$. Such an $h$ carries orbits of $A$ to orbits of $B$, so topologically conjugate systems are indistinguishable by every purely dynamical, coordinate-free property; the classification problem asks for a usable list of invariants deciding exactly when two matrices are conjugate in this sense. It is the discrete-time counterpart of the topological classification of linear flows (continuous-time systems $\dot{x}=Ax$), which Berger and Wynne completed in the source paper.

The problem splits sharply by spectrum. Kuiper and Robbin (1973) proved that when the (complex) eigenvalues of a real linear operator include no root of unity (a complex number $\lambda$ with $\lambda^q=1$ for some positive integer $q$), topological conjugacy is equivalent to an elementary finite list of data: the dimensions and orientations of the expanding and contracting spectral summands, together with the ordinary linear-conjugacy class of the restriction to the summand whose eigenvalues lie on the unit circle. They also showed that the general case reduces to the periodic case — operators of finite order, whose eigenvalues are all roots of unity — and conjectured that topologically conjugate finite-order operators must be linearly similar, verifying this for orders $1,2,3,4,6$. That conjecture is false in general: Cappell and Shaneson, using surgery theory (a high-dimensional topology technique for modifying and classifying manifolds via cut-and-paste), constructed the first examples of nonlinear similarity — topological conjugacy without linear similarity — and with Steinberger and West showed such examples exist precisely in dimensions six and above, while in dimensions at most five topological conjugacy of linear operators implies linear similarity. Hsiang and Pardon proved by related methods that topological equivalence forces linear equivalence for orthogonal transformations generating a cyclic group of odd order or twice odd order. Cruz (1991) showed the Lipschitz analogue is rigid: Lipschitz-conjugate linear operators are linearly similar, confirming that all difficulty is concentrated at $C^0$. Hambleton and Pedersen (2005) completed the torsion-theoretic analysis for finite cyclic groups, showing that nonlinear similarity is detected by Reidemeister torsion (a numerical invariant, defined up to units, attached to a chain complex with a basis) together with number-theoretic congruences; their criterion is stated modulo the image of a relative assembly map and is made fully effective only for cyclic groups of two-power order, leaving the general periodic case without a complete decision procedure. Subsequent work in the Sergeichuk school repeatedly restates the problem as open, and Luo (2021) gave a constructive dynamical classification for complex matrices that deliberately leaves the roots-of-unity regime, and the classical Kuiper–Robbin conjecture, unresolved. The source authors themselves, in later papers on Hölder and Lipschitz classifications of linear flows (2025–2026), still describe the discrete-time $C^0$ operator problem as significantly more challenging than the continuous-time one.

Problem Statement

Give a complete classification of linear operators on finite-dimensional vector spaces up to topological ($C^0$) conjugacy: for linear maps $A:X\to X$ and $B:Y\to Y$ over the real (and, correspondingly, the complex) ground field, provide necessary and sufficient conditions — expressed through effectively computable data attached to $A$ and $B$ — under which there exists a homeomorphism $h:X\to Y$ with $h(Ax)=Bh(x)$ for all $x$, so that any two operators can algorithmically be decided to be topologically conjugate or not, and the conjugacy class of each operator can be read off. By the Kuiper–Robbin reduction this is solved for operators whose spectrum contains no root of unity, and the remaining open core is the roots-of-unity (periodic) regime: complete the classification of finite-order linear maps up to topological conjugacy — equivalently, the nonlinear-similarity classification of finite cyclic group representations beyond the currently settled sub-cases (dimensions at most five, odd and twice-odd cyclic order, two-power cyclic order) — and assemble with the known non-roots-of-unity criteria a decision procedure for the general operator, including non-diagonalizable (Jordan) structure at roots of unity.

References

1. Arno Berger and Anthony Wynne, "On the Classification of Finite-Dimensional Linear Flows", Journal of Dynamics and Differential Equations 32 (2020) 23–59, doi:10.1007/s10884-018-9717-4; preprint arXiv:1806.03385, https://arxiv.org/abs/1806.03385 (source of the open question)
2. Nicolaas H. Kuiper and Joel W. Robbin, "Topological classification of linear endomorphisms", Inventiones Mathematicae 19 (1973) 83–106, doi:10.1007/BF01418922, https://doi.org/10.1007/BF01418922
3. Nicolaas H. Kuiper and Joel W. Robbin, "Topological classification of linear endomorphisms" (companion note), Annales de l'Institut Fourier 23(2) (1973) 93–95, doi:10.5802/aif.459, https://doi.org/10.5802/aif.459
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15. Arno Berger and Anthony Wynne, "Hölder classifications of finite-dimensional linear flows", preprint arXiv:2511.02001 (2025), https://arxiv.org/abs/2511.02001
16. Arno Berger and Anthony Wynne, "On Lipschitz equivalence of finite-dimensional linear flows", preprint arXiv:2602.13877 (2026), https://arxiv.org/abs/2602.13877