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Section 1 Introduction

A Carnot group is a simply connected nilpotent Lie group \(G\) whose Lie algebra \(\mathfrak{g}\) admits a stratification \(\mathfrak{g}=\lielayer{\mathfrak{g}}{1}\oplus\dots\oplus\lielayer{\mathfrak{g}}{s}\text{,}\) i.e., a decomposition such that \([\lielayer{\mathfrak{g}}{1}, \lielayer{\mathfrak{g}}{i}] = \lielayer{\mathfrak{g}}{i+1}\) for all \(i=1,\ldots,s\text{,}\) where \(\lielayer{\mathfrak{g}}{s+1}=\{0\}\text{.}\) The dimension \(r=\dim\lielayer{\mathfrak{g}}{1}\) is called the rank of the Carnot group \(G\) and the largest integer \(s\) such that \(\lielayer{\mathfrak{g}}{s}\neq\{0\}\) is called the step. Let

\begin{equation*} \operatorname{End}\colon L^2([0,1];\lielayer{\mathfrak{g}}{1}) \to G,\quad u\mapsto \gamma_u(1) \end{equation*}

be the endpoint map, where \(\gamma_u\colon [0,1]\to G\) is the horizontal curve starting from the identity element \(\identity{G}\in G\) with control \(u\text{,}\) i.e., the unique absolutely continuous curve such that

\begin{equation*} \frac{d}{dt}\gamma_u(t) = (L_{\gamma(t)})_*u(t)\quad\text{and}\quad \gamma_u(0) = \identity{G}\text{.} \end{equation*}

Here \(L_{\gamma(t)}\colon G\to G\) is the left translation \(L_{\gamma(t)}(g) = \gamma(t)\cdot g\).

The abnormal curves are the trajectories \(\gamma_u\) for critical points \(u\in L^2([0,1];\lielayer{\mathfrak{g}}{1})\) of the endpoint map. Their significance arises from the Pontryagin Maximum Principle, which separates potentially length-minimizing curves in sub-Riemannian manifolds into two types: the normal and the abnormal extremals, see e.g. Section 3.4 of [1] for details. The normal extremals are well behaved, but the abnormal ones are the subject of two important open problems of sub-Riemannian geometry: the regularity of length minimizers and the Sard Conjecture, see Section 10 of [21].

The regularity problem asks what is the minimal regularity of sub-Riemannian length-minimizing curves. The issue is the existence of strictly abnormal minimizers, see [20], [19], [10] for some examples. The abnormal curves in general have no regularity beyond being Lipschitz, but nonetheless all known examples of abnormal length minimizers are \(C^\infty\)-smooth. Many partial regularity results for length minimizers exist, such as \(C^\infty\)-regularity in generic sub-Riemannian structures of rank at least \(3\) [9], analyticity on an open dense subset of each minimizer [24], nonexistence of corner type singularities [13], and \(C^1\)-regularity in dimension 3 [4].

The Sard Conjecture is that the set of critical values of the endpoint map should have zero measure, i.e., most points should not be reachable from a fixed initial point with an abnormal curve. The set of critical values is known as the abnormal set. A more restricted variant is the Minimizing Sard Conjecture, where the abnormal curves are in addition required to be length minimizers. As with the regularity problem, several partial results exist. For instance, the minimizing abnormal set is contained in a closed nowhere dense set [2], the abnormal set is a proper algebraic or analytic subvariety in Carnot groups and polarized groups of particular types [17], and in dimension 3 the abnormal set is a countable union of semianalytic curves [4].

Recently, both the regularity and Sard problems have seen progress through the study of abnormal curves from a dynamical systems viewpoint. A class of potentially minimizing abnormal curves in rank 2 sub-Riemannian structures was proved to have at least \(C^1\)-regularity [3], and the Sard Conjecture was proved in Carnot groups of rank 2 step 4 and rank 3 step 3 [8]. The idea common to both articles is that differentiating the identities defining abnormal curves leads to an ODE system that some reparametrization of the control of the abnormal curve will satisfy. Then using normal forms for the resulting ODEs, the authors of [3] and [8] were able to prove their respective claims by studying trajectories of finitely many explicit ODE systems.

One reason for the success of the previous two articles was that the dynamical systems were not completely arbitrary, but had some properties that helped simplify the systems, such as the linear part being a traceless matrix. The goal of this paper is to study the scope of these dynamical systems within the setting of Carnot groups. The underlying questions motivating this research are:

  • Which dynamical systems can arise as the ODE systems of abnormal curves in Carnot groups?
  • Which curves in \(\RR^r\) are the horizontal projections of abnormal curves in Carnot groups of rank \(r\text{?}\)

The main result of this paper is that without fixing a specific Carnot group to study, there are essentially no restrictions on the possible dynamics of abnormal trajectories.

The Carnot group \(F\) in Theorem 1.1 may be taken to be a free Carnot group whose step depends only on the degrees of the polynomials of the ODE \(P\text{,}\) see Theorem 4.1 for the more precise version. Applying Theorem 1.1 to the simplest possible Carnot groups \(G=\RR^r\) shows that all trajectories of polynomial ODEs in \(\RR^r\) are the horizontal projections of abnormal curves of Carnot groups of rank \(r\text{.}\) Several examples of this type will be presented in Section 5. Theorem 1.1 can also be viewed as an algebraic complexity counterpart to the metric complexity result of [18] that arbitrary curves in \(\RR^r\) are well approximated by horizontal projections of length minimizing curves of rank \(r\) Carnot groups.

The ingredients used to prove Theorem 1.1 also produce an auxiliary result that finite concatenations of abnormal curves have abnormal lifts.

A potentially useful feature of the method of Theorem 1.1 and Theorem 1.2 is that the abnormal lifts satisfy the Goh condition, see e.g. Section 12.3 of [1] for the definition. Moreover the groups \(F\) can be chosen such that the abnormal lifts also satisfy even higher order abnormality conditions such as the third order condition of [5].

Subsection 1.1 Structure of the paper

Section 2 lays the foundations needed to prove the main results. The characterization of abnormal curves in terms of abnormal polynomials is covered in Subsection 2.1. Subsection 2.2 recalls the Hall basis construction for free Lie algebras and fixes the particular basis to be used in the rest of the paper. Subsection 2.3 describes how to use the fixed Hall basis to transfer abnormal polynomials between Carnot groups of different step by treating them as polynomials in a free Lie algebra. An important construction is the action of a free Lie algebra as derivations on its polynomial ring. Finally, Subsection 2.4 recalls the definition of a Poincaré series to facilitate counting the number of monomials of given degrees.

The core of the proofs of the main theorems is covered in Section 3. Subsection 3.1 reduces abnormality of a trajectory of an ODE to the existence of a common factor in abnormal polynomials. Subsection 3.2 defines a quotient Lie algebra that eliminates irrelevant variables from the abnormal polynomials. The key argument is in Subsection 3.3, which rephrases finding a common factor as an eventually underdetermined linear system. The proofs of the main theorems are concluded in Section 4.

The final part of the paper in Section 5 is dedicated to constructing examples of abnormal curves. The proof of Theorem 1.1 is condensed into an algorithm and several examples are computed, showcasing some possibilities and limitations of the algorithm.