Dynamics on character varieties: Brown’s theorem revisited

Let be a surface of genus with punctures. Given a Lie group , the -character variety of is the space of representations modulo conjugations by elements of .

The mapping class group of isotopy classes of orientation-preserving diffeomorphisms of acts naturally on .

The dynamics of mapping class groups on character varieties was systematically studied by Goldman in 1997: in his landmark paper, he showed that the -action on is ergodic with respect to Goldman–Huebschmann measure whenever .

Remark 1 This nomenclature is not standard: we use it here because Goldman showed here that has a volume form coming from a natural symplectic structure and Huebschmann proved here that this volume form has finite mass.

The ergodicity result above partly motivates the question of understanding the dynamics of individual elements of mapping class groups acting on -character varieties.

In this direction, Brown studied in 1998 the actions of elements of on the character variety . As it turns out, if is a small loop around the puncture, then the -action on preserves each level set , , of the function sending to the trace of the matrix . Here, Brown noticed that the dynamics of elements of on level sets with close to fit the setting of the celebrated KAM theory (assuring the stability of non-degenerate elliptic periodic points of smooth area-preserving maps). In particular, Brown tried to employ Moser’s twisting theorem to conclude that no element of can act ergodically on all level sets , .

Strictly speaking, Brown’s original argument is not complete because Moser’s theorem is used without checking the twist condition.

In the sequel, we revisit Brown’s work in order to show that his conclusions can be derived once one replaces Moser’s twisting theorem by a KAM stability theorem from 2002 due to Rüssmann.

1. Statement of Brown’s theorem

1.1. -character variety of a punctured torus

Recall that the fundamental group of an once-punctured torus is naturally isomorphic to a free group on two generators and such that the commutator corresponds to a loop around the puncture of .

Therefore, a representation is determined by a pair of matrices , and an element of the -character variety of is determined by the simultaneous conjugacy class , , of a pair of matrices .

The traces , and of the matrices , and provide an useful system of coordinates on : algebraically, this is an incarnation of the fact that the ring of invariants of is freely generated by the traces of , and .

In particular, the following proposition expresses the trace of in terms of , and .

Proposition 1 Given , one has

Proof: By Cayley–Hamilton theorem (or a direct calculation), any satisfies , i.e., .

Hence, for any , one has

so that

It follows that, for any , one has

and

Since and , the proof of the proposition is complete.

1.2. Basic dynamics of on character varieties

Recall that the mapping class group is generated by Dehn twists and about the generators and of . In appropriate coordinates on the once-punctured torus , the isotopy classes of these Dehn twists are represented by the actions of the matrices

on the flat torus . In particular, at the homotopy level, the actions of and on are given by the Nielsen transformations

Since the elements of fix the puncture of , they preserve the homotopy class of a small loop around the puncture. Therefore, the -action on the character variety respects the level sets , , of the function given by

Furthermore, each level set , , carries a finite (Goldman—Huebschmann) measure coming from a natural -invariant symplectic structure.

In this context, the level set corresponds to impose the restriction , so that is naturally identified with the character variety .

In terms of the coordinates , and on , we can use Proposition 1 (and its proof) and (1) to check that

and

Hence, we see from (2) that:

• the level set consists of a single point ;
• the level sets , , are diffeomorphic to -spheres;
• the character variety is a -dimensional orbifold whose boundary is a topological sphere with 4 singular points (of coordinates with ) corresponding to the character variety .

After this brief discussion of some geometrical aspects of , we are ready to begin the study of the dynamics of . For this sake, recall that the elements of are classified into three types:

• is called elliptic whenever ;
• is called parabolic whenever ;
• is hyperbolic whenever .

The elliptic elements have finite order (because and ) and the parabolic elements are conjugated to for some .

In particular, if is elliptic, then leaves invariant non-trivial open subsets of each level set , . Moreover, if is parabolic, then preserves a non-trivial and non-peripheral element and, a fortiori, preserves the level sets of the function , . Since any such function has a non-constant restriction to any level set , , Brown concluded that:

Proposition 2 (Proposition 4.3 of Brown’s paper) If is not hyperbolic, then its action on is not ergodic whenever .

On the other hand, Brown observed that the action of any hyperbolic element of on can be understood via a result of Katok.

Proposition 3 (Theorem 4.1 of Brown’s paper) Any hyperbolic element of acts ergodically on .

Proof: The level set is the character variety . In other words, a point in represents the simultaneous conjugacy class of a pair of commuting matrices in .

Since a maximal torus of is a conjugate of the subgroup

we have that is the set of simultaneous conjugacy classes of elements of . In view of the action by conjugation

of the element of the Weyl subgroup of , we have

In terms of the coordinates given by the phases of the elements

the element acts by , so that is the topological sphere obtained from the quotient of by its hyperelliptic involution (and has only four singular points located at the subset of fixed points of the hyperelliptic involution). Moreover, an element acts on by mapping to .

In summary, the action of on is given by the usual -action on the topological sphere induced from the standard on the torus .

By a result of Katok, it follows that the action of any hyperbolic element of on is ergodic (and actually Bernoulli).

1.3. Brown’s theorem

The previous two propositions raise the question of the ergodicity of the action of hyperbolic elements of on the level sets , . The following theorem of Brown provides an answer to this question:

Theorem 4 Let be an hyperbolic element of . Then, there exists such that does not act ergodically on .

Very roughly speaking, Brown establishes Theorem 4 along the following lines. One starts by performing a blowup at the origin in order to think of the action of on as a one-parameter family , , of area-preserving maps of the -sphere such that is a finite order element of . In this way, we have that is a non-trivial one-parameter family going from a completely elliptic behaviour at to a non-uniformly hyperbolic behaviour at . This scenario suggests that the conclusion of Theorem 4 can be derived via KAM theory in the elliptic regime.

In the next (and last) section of this post, we revisit Brown’s ideas leading to Theorem 4 (with an special emphasis on its KAM theoretical aspects).

2. Revisited proof of Brown’s theorem

2.1. Blowup of the origin

The origin of the character variety can be blown up into a sphere of directions . The action of on factors through an octahedral subgroup of : this follows from the fact that (3) implies that the generators and of act on as

In this way, each element is related to a root of unity

of order coming from the eigenvalues of the derivative of at any of its fixed points.

Example 1 The hyperbolic element acts on via the element of of order .

2.2. Bifurcations of fixed points

An hyperbolic element induces a non-trivial polynomial automorphism of whose restriction to describe the action of on . In particular, the set of fixed points of this polynomial automorphism in is a semi-algebraic set of dimension .

Actually, it is not hard to exploit the fact that acts on the level sets , , through area-preserving maps to compute the Zariski tangent space to in order to verify that is one-dimensional (cf. Proposition 5.1 in Brown’s work).

Moreover, this calculation of Zariski tangent space can be combined with the fact that any hyperbolic element has a discrete set of fixed points in and, a fortiori, in to get that is transverse to except at its discrete subset of singular points and, hence, is discrete for all (cf. Proposition 5.2 in Brown’s work).

Example 2 The hyperbolic element acts on via the polynomial automorphism (cf. (3)). Thus, the corresponding set of fixed points is given by the equations

describing an embedded curve in .

In general, the eigenvalues of the derivative at of the action of an hyperbolic element on can be continuously followed along any irreducible component of .

Furthermore, it is not hard to check that is not constant on (cf. Lemma 5.3 in Brown’s work). Indeed, this happens because there are only two cases: the first possibility is that connects and so that varies from to the unstable eigenvalue of acting on ; the second possibility is that becomes tangent to for some so that the Zariski tangent space computation mentioned above reveals that varies from (at ) to some value (at any point of transverse intersection between and a level set of ).

2.3. Detecting Brjuno elliptic periodic points

The discussion of the previous two subsections allows to show that the some portions of the action of an hyperbolic element fit the assumptions of KAM theory.

Before entering into this matter, recall that is Brjuno whenever is an irrational number whose continued fraction has partial convergents satisfying

For our purposes, it is important to note that the Brjuno condition has full Lebesgue measure on .

Let be an hyperbolic element. We have three possibilities for the limiting eigenvalue : it is not real, it equals or it equals .

If the limiting eigenvalue is not real, then we take an irreducible component intersecting the origin . Since is not constant on implies that contains an open subset of . Thus, we can find some such that has a Brjuno eigenvalue , i.e., the action of on has a Brjuno fixed point.

If the limiting eigenvalue is , we use Lefschetz fixed point theorem on the sphere with close to to locate an irreducible component of such that is a fixed point of positive index of for close to . On the other hand, it is known that an isolated fixed point of an orientation-preserving surface homeomorphism which preserves area has index . Therefore, is a fixed point of of index with multipliers close to whenever is close to . Since a hyperbolic fixed point with positive multipliers has index , it follows that is a fixed point with when is close to . In particular, contains an open subset of and, hence, we can find some such that has a Brjuno multiplier .

If the limiting eigenvalue is , then is an hyperbolic element with limiting eigenvalue . From the previous paragraph, it follows that we can find some such that contains a Brjuno elliptic fixed point of .

In any event, the arguments above give the following result (cf. Theorem 4.4 in Brown’s work):

Theorem 5 Let be an hyperbolic element. Then, there exists such that has a periodic point of period one or two with a Brjuno multiplier.

2.4. Moser’s twisting theorem and Rüssmann’s stability theorem

At this point, the idea to derive Theorem 4 is to combine Theorem 5 with KAM theory ensuring the stability of certain types of elliptic periodic points.

Recall that a periodic point is called stable whenever there are arbitrarily small neighborhoods of its orbit which are invariant. In particular, the presence of a stable periodic point implies the non-ergodicity of an area-preserving map.

A famous stability criterion for fixed points of area-preserving maps is Moser’s twisting theorem. This result can be stated as follows. Suppose that is an area-preserving , , map having an elliptic fixed point at origin with multipliers , such that for . After performing an appropriate area-preserving change of variables (tangent to the identity at the origin), one can bring into its Birkhoff normal form, i.e., has the form

where , , are uniquely determined Birkhoff constants and denotes higher order terms.

Theorem 6 (Moser twisting theorem) Let be an area-preserving map as in the previous paragraph. If for some , then the origin is a stable fixed point.

The nomenclature “twisting” comes from the fact when is a twist map, i.e., has the form in polar coordinates where is a smooth function with . In the literature, the condition “ for some ” is called twist condition.

Example 3 The Dehn twist induces the polynomial automorphism on . Each level set , , is a smooth -sphere which is swept out by the -invariant ellipses obtained from the intersections between and the planes of the form .Goldman observed that, after an appropriate change of coordinates, each becomes a circle where acts as a rotation by angle . In particular, the restriction of to each level set is a twist map near its fixed points .

In his original argument, Brown deduced Theorem 4 from (a weaker version of) Theorem 5 and Moser’s twisting theorem. However, Brown employed Moser’s theorem with while checking only the conditions on the multipliers of the elliptic fixed point but not the twist condition .

As it turns out, it is not obvious to check the twist condition in Brown’s setting (especially because it is not satisfied at the sphere of directions ).

Fortunately, Rüssmann discovered that a Brjuno elliptic fixed point of a real-analytic area-preserving map is always stable (independently of twisting conditions):

Theorem 7 (Rüssmann) Any Brjuno elliptic periodic point of a real-analytic area-preserving map is stable.

Remark 2 Actually, Rüssmann obtained the previous result by showing that a real-analytic area-preserving map with a Brjuno elliptic fixed point and vanishing Birkhoff constants (i.e., for all ) is analytically linearisable. Note that the analogue of this statement in the category is false (as a counterexample is given by ).

In any case, at this stage, the proof of Theorem 4 is complete: it suffices to put together Theorems 5 and 7.