Topological conjugacy of linear endomorphisms: the Kuiper–Robbin classification problem, its root-of-unity core, and discrete-time real projective transformations

Two linear endomorphisms $f, g$ of $\mathbb{R}^n$ are topologically conjugate when a homeomorphism $h$ of $\mathbb{R}^n$ satisfies $h \circ f = g \circ h$. In 1973 Kuiper and Robbin posed the problem of classifying all linear endomorphisms of finite-dimensional real vector spaces up to this equivalence, proposed an explicit criterion (linear conjugacy of the unit-circle parts together with matching dimension and orientation data of the expanding and contracting parts), and proved that their criterion is valid whenever no eigenvalue of the maps involved is a root of unity (apart from a short list of small exceptional values in their statement). Their conjecture that linear data would also suffice in general is now known to be false: topologically similar but linearly inequivalent finite-order maps exist from dimension six upward (Cappell–Shaneson), while in dimensions at most 5 topological and linear similarity coincide. That root-of-unity core coincides with the 'nonlinear similarity' program of Cappell–Shaneson, Hsiang–Pardon, Schultz, Hambleton–Pedersen and others; Hambleton and Pedersen (2005) proved that Reidemeister torsion and congruences in representation weights determine nonlinear similarity for finite cyclic groups, with a complete classification for cyclic groups of 2-power order, but no assembled classification of linear endomorphisms in full generality is published, and applied classification papers as late as 2014 still explicitly exclude operators having a root-of-unity eigenvalue. A second open component is the topological conjugacy classification of discrete-time real projective transformations, for which only Kuiper's partial 1976 results (with further partial results by Gopal–Ravulapalli 2018) are available, in contrast with the completed continuous-time projective case. The two components together ask for the still-missing topological conjugacy classification of linear dynamical systems in discrete time.

Background

A discrete-time linear dynamical system is given by iterating a linear endomorphism $f: \mathbb{R}^n \to \mathbb{R}^n$. The natural notion of sameness for such systems is topological conjugacy: $f \sim g$ if there exists a homeomorphism $h$ (not assumed linear) with $h \circ f = g \circ h$; such an $h$ carries orbits of $f$ to orbits of $g$ and preserves all purely dynamical, coordinate-free behavior. Linear conjugacy ($h$ linear invertible) is decided exactly by real Jordan form: it matches the multiset of Jordan blocks for each real eigenvalue and each conjugate pair of complex eigenvalues. Every endomorphism decomposes $\mathbb{R}^n = E^+ \oplus E^0 \oplus E^-$ into a direct sum of invariant subspaces on which all eigenvalues have modulus $> 1$ (the expanding part), $= 1$ (the unit-circle part $f_0$), or $< 1$ (the contracting part); $E^+$ and $E^-$ carry orientation data determined by $f$ (whether $f$ preserves or reverses orientation on each of these two summands), and Kuiper–Robbin's proposed invariants are the linear-conjugacy class of $f_0$ together with the dimensions and orientation data of the expanding and contracting parts. The hard residual case concerns eigenvalues on the unit circle that are roots of unity, i.e. $\lambda$ with $\lambda^q = 1$ for some $q \geq 1$: when such eigenvalues are present the conjugacy problem becomes entangled with the nonlinear similarity problem for the finite cyclic group $\mathbb{Z}_q$. There, two orthogonal representations of a finite group (i.e. actions of the group by orthogonal linear maps on a Euclidean space) are linearly similar if they are conjugate by a linear map and topologically similar (nonlinearly similar) if they are conjugate by a homeomorphism; de Rham conjectured that topological similarity should imply linear equivalence and showed that Whitehead–Reidemeister torsion, an algebraic invariant from chain complexes that is preserved by topological similarity but can distinguish linear equivalence classes, governs many cases. Cappell and Shaneson subsequently constructed, by methods of surgery theory (the machinery classifying high-dimensional manifolds up to homeomorphism), topologically similar representations over $\mathbb{Z}_{4q}$ that are not linearly similar in dimensions above 6, so the two equivalence relations genuinely differ there (and the finite-order case generates the first examples in dimension six); in dimensions at most 5, Cappell and Shaneson (1999) proved that topological and linear similarity coincide. Hambleton and Pedersen (2005) then completed de Rham's program for cyclic groups, proving that Reidemeister torsion together with explicit congruences in the weights of the free representation summands determines nonlinear similarity for finite cyclic groups, with a complete classification for cyclic groups of 2-power order; for general cyclic orders their criterion retains an algebraically described indeterminacy, and no published work assembles these results into an explicit topological-conjugacy classification of arbitrary linear endomorphisms. The second component concerns real projective space $\mathbb{RP}^{n-1}$, the space of lines through the origin in $\mathbb{R}^n$, on which each invertible linear map induces a projective transformation; classifying projective transformations up to topological conjugacy is the quotient-level analogue of the linear problem, and while the continuous-time (flow) version was completely classified by Ayala and Kawan in 2014 building on Kuiper and Ladis, the discrete-time version is settled only in the partial cases Kuiper established in 1976.

Problem Statement

(1) Let $f$ and $g$ be linear endomorphisms of a finite-dimensional real vector space. Give a complete classification of such endomorphisms up to topological conjugacy: necessary and sufficient conditions for the existence of a homeomorphism $h$ with $h \circ f = g \circ h$. Kuiper and Robbin settled the case in which no eigenvalue is a root of unity: then $f$ and $g$ are topologically conjugate if and only if their unit-circle parts (the invariant summands on which all eigenvalues have modulus 1) are linearly conjugate and the expanding and contracting summands agree in dimension and orientation. Their conjecture that the same criterion holds in general — equivalently, that two finite-order automorphisms are topologically conjugate if and only if linearly conjugate, the statement about finite-order elements of the orthogonal group $O(n+1,\mathbb{R})$ acting on the unit sphere $S^n$ — is refuted: topologically similar but linearly inequivalent finite-order maps exist from dimension six upward (Cappell–Shaneson), while in dimensions at most 5 topological and linear similarity coincide. The finite-order core is equivalent to the nonlinear similarity problem for representations of finite cyclic groups $\mathbb{Z}_q$, for which Hambleton and Pedersen (2005) proved that Reidemeister torsion and congruences in representation weights determine similarity, with a complete classification for cyclic groups of 2-power order. No complete topological-conjugacy classification of linear endomorphisms is known. (2) Classify real projective transformations — the self-maps of real projective space $\mathbb{RP}^{m}$ induced by invertible linear maps of $\mathbb{R}^{m+1}$ — up to topological conjugacy in discrete time, beyond the partial cases established by Kuiper (1976) and the further partial results of Gopal–Ravulapalli (2018); the continuous-time analogue (real projective flows) was completely classified by Ayala and Kawan (2014).

References

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