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pp. 979–984

DYNAMICAL SYSTEMS

Volume 9, Number 4, July 2003

DIFFERENTIABILITY OF THE HARTMAN–GROBMAN

LINEARIZATION

MISHA GUYSINSKY, BORIS HASSELBLATT, AND VICTORIA RAYSKIN

(Communicated by Anatole Katok)

Abstract. We show that the linearizing homeomorphism in the Hartman–

Grobman Theorem is diﬀerentiable at the ﬁxed point.

1. Introduction. The well-known theorem of Grobman and Hartman is a basic

example of local analysis in hyperbolic dynamical systems:

Theorem 1 (Hartman, Grobman [G, H2]). Let U ⊂ Rn be a neighborhood of 0,

f : U → Rn continuously diﬀerentiable with 0 as a hyperbolic ﬁxed point. Then

there is a homeomorphism h of a neighborhood of 0 with h ◦ f = Df0 ◦ h near 0.

Thus, the dynamics near the hyperbolic ﬁxed point is topologically the same

as that of the linear part of the map. This helps piece together phase portraits

by inserting a small linearized phase portrait around every hyperbolic ﬁxed point,

which one can then hope to plausibly connect to create the whole phase portrait.

If this is expected to be not just a topological rendering of the dynamics, however,

but to distinguish diﬀerent kinds of nodes, then a topological coordinate change

is not suﬃcient, and this has motivated research into the question of whether the

regularity of the conjugating homeomorphism can be improved.

Hartman showed early on that in dimension 2 (or under some spectral conditions)

the conjugating map is C1 [H1], but gave an example in the same paper to show

that this can fail in dimension 3. Yet, examination of a proof of the Hartman–

Grobman Theorem (e.g., in [KH]) as well as analogy [KH, Section 19.1] suggests

that the conjugacy is Ho¨lder continuous. Indeed, van Strien [S] asserts that this is

well-known. An published proof turned out a little hard to ﬁnd, but a preprint by

Belitski˘ı contains the desired statement:

Theorem 2 ([B]). The linearization in the Hartman–Grobman Theorem is α-

H¨older (and α can be given in terms of contraction and expansion rates).

Unless, however, the Ho¨lder exponent can be chosen arbitrarily near 1, there is

no a priori guarantee that diﬀerent nodes will be distinguished reliably. Also, one

would like to establish the eigendirections as meaningful for the nonlinear map.

Accordingly, the purpose of the present paper is to establish the following result:

Theorem 3. If the map f in the Hartman–Grobman Theorem is C∞ then the

linearizing homeomorphism is diﬀerentiable at the origin, and its derivative at 0 is

the identity.

979

980 MISHA GUYSINSKY, BORIS HASSELBLATT, AND VICTORIA RAYSKIN

We believe that this result holds for f ∈ C2, as announced by van Strien [S], but

his approach runs into diﬃculties (such as the failure of his Proposition 4.1) that

appear hard to overcome. We use a diﬀerent approach inspired from normal forms

that is easier to execute. The only price is the C∞ requirement, which enters in

the use of Theorem 4. It suﬃces to require ﬁnite diﬀerentiability, but of an order

that depends on spectral information in a slightly involved way.

2. Outline of the proof of Theorem 3. We wish to show that the conjugacy

is within o(x) of the identity. The ﬁrst step is to invoke a result of Bronstein and

Kopanski˘ı in order to reduce the map to its second-order normal form expansion

via a C1 conjugacy. The iterates of this map are polynomials, and we estimate

their coeﬃcients in Section 3. In Section 5 we deﬁne the conjugacy via a natural

correspondence between orbits under the linear and nonlinear dynamics. We com-

bine an estimate as to how close to the origin a point must be in order to stay in

a given neighborhood for a speciﬁed time (Section 4) with the coeﬃcient estimates

to control the diﬀerence between orbits for the linear and nonlinear maps.

3. Coeﬃcient estimates. We use a result of Bronstein and Kopanski˘ı [BK]:

Theorem 4. If f : Rd → Rd is a C∞ diﬀeomorphism such that f (0) = 0 and

L := Df (0) is hyperbolic then on a neighborhood of 0 there is a C1 conjugacy H+ =

Id +∆h with ∆h(x) = o( x ) between f and a quadratic polynomial f(x) := Lx +

Q(x) that contains only weakly resonant terms as explained in the proof of Lemma 7.

Remark 5. Note in particular that f(0) = 0 and the stable and unstable manifolds

of 0 under f coincide with those under L.

Proof. This follows from [BK, Theorem 11.9] once one veriﬁes that monomials

of order greater than 2 satisfy the condition A(1). Per [BK, Remark 7.6], A(1)

follows from the condition S(1), and this is relatively straightforward to check [BK,

bottom p. 191]. Indeed, denote by −λl < · · · < −λ1 < 0 < µ1 < · · · < µm

the distinct values of log |ν|, where ν is an eigenvalue of L. For a multiindex

τ = (αl, . . . , α1, β1, . . . , βm) the condition S(1) [BK, p. 111], is that

rs

either αiλi > λr for some r ≤ l or βjµj > µs for some s ≤ m.

i=1 j=1

For terms of order at least 3 the multiindex-exponent satisﬁes |τ |:= αi+ βj ≥ 3

and hence S(1) because either αi ≥ 2, in which case r = max{i | αi > 0} works,

or else s = max{j | βj > 0} does.

Remark 6. This is the only place where we use the C∞ assumption, and, as in

[BK, Theorem 11.9] it can be replaced by a Ck assumption, where k depends on the

spectrum of the linear part. As this dependence is a little complicated we assume

C∞, but the reader may study the precise assumptions of [BK, Theorem 11.9].

Consider the root space decomposition of L := Df (0) and deﬁne a linear map

D to be a scaling by a on each root space for an eigenvalue of absolute value a.

Deﬁne J by L = DJ and note (for example via the Jordan normal form of L) that

DJ = JD, all eigenvalues of J are 1, and that the entries of Jn are polynomials in

n. We assume our coordinate system is adaped to the root space decomposition,

so J is block diagonal. We occasionally write Q(x) for Q(x, x).

Lemma 7. Q ◦ D = DQ.

DIFFERENTIABILITY OF THE HARTMAN–GROBMAN LINEARIZATION 981

Proof. This restates that Q contains only weakly resonant terms according to The-

orem 4. By deﬁnition, this means that if (in coordinates adapted to the root space

decomposition) the ith component of Q contains a term a · xy with a = 0 then the

eigenvalues λ and µ associated with the root spaces corresponding to x and y, re-

spectively, are related to the eigenvalue η associated with the root space containing

the ith unit vector by |η| = |λ||µ|. Note that this is exactly the claim.

Remark 8. The conjugacy is related to hn := L−nfn, where fn =: Ln(Id + NLn),

and accordingly, we wish to estimate the coeﬃcients of its nonlinear terms. To that

end denote by bn,m the sum of the absolute values of all mth-order coeﬃcients in

the coordinate representation of hn (this is the height of the mth-order term of hn).

Lemma 9. There are N, k ∈ N such that bn+1,m ≤ bn,m +N nk−1 m−1 bn,p bn,m−p

p=1

for n, m ∈ N.

Proof. Id + NLn+1 = hn+1 = L−n−1f ◦ fn = L−n−1(L + Q)Ln(Id + NLn), so

NLn+1 = NLn + L−n−1Q( Ln(Id + NLn))

= NLn + D−n−1J −n−1Q(DnJ n(Id + NLn)) (1)

= NLn + D−1J −n−1Q( J n(Id + NLn)).

Here we used Lemma 7 and DJ = JD.

To bound bn+1,m note ﬁrst that terms of a given order m in a coordinate repre-

sentation of Jn(Id + NLn) come in linear combinations with polynomial coeﬃcients

of mth order terms in Id + NLn. These polynomial coeﬃcients arise from entries of

Jn, and the form of the linear combinations is otherwise independent of n. Thus,

the sum βn,p of the absolute values of all pth-order coeﬃcients in the coordinate

representation of Jn(Id + NLn) is at most P1(n)bn,p for some polynomial P1 that

is independent of p because it encodes only the action of Jn.

Likewise, in a coordinate representation of D−1J−n−1Q(Jn(Id + NLn)) terms of

a given order m come in linear combinations with polynomial coeﬃcients of mth

order terms in Q(Jn(Id + NLn)). These coeﬃcients arise from entries of J−n−1,

and the form of the linear combinations is otherwise independent of n. Thus, the

sum of mth-order coeﬃcients in D−1J−n−1Q(Jn(Id + NLn)) is bounded in terms

of that in Q(Jn(Id + NLn)) by including a polynomial multiplier P2(n).

Sorting by the order m of terms, the ith component of Q(Jn(Id + NLn)) is

m−1

aijl[J n(Id + NLn)]j [J n(Id + NLn)]l = aijl αj,n,ρxρ αl,n,τ xτ.

j,l m p=1 j,l |ρ|=p |τ |=m−p

Thus, the absolute values of all mth order coeﬃcients in Q(Jn(Id + NLn)) sum to

m−1

|aijl|[ |αj,n,ρ|][ |αl,n,τ |] ≤ c0 βn,pβn,m−p,

i p=1 j,l |ρ|=p |τ |=m−p p

using |ρ|=p,|τ |=m−p |αj,n,ραl,n,τ | = |ρ|=p |αj,n,ρ| |τ |=m−p |αl,n,τ |. Take N, k ∈

N such that P2(n)c0P12(n) ≤ N nk−1 for all n ∈ N to get the claim by (1).

This recursion relation allows us to bound the coeﬃcients inductively.

Lemma 10. For any α > 0 there exists C > 0 such that bn,m ≤ eα(n+C)m.

982 MISHA GUYSINSKY, BORIS HASSELBLATT, AND VICTORIA RAYSKIN

Proof. If αC ≥ supn 2 log N +2k log n−αn then (N 2n2k)m ≤ eα(n+C)m, so it suﬃces

to show inductively that if N, k are as in Lemma 9 then bn,m ≤ (N nk)2m−1. This is

clear for n = 1, and if it is known for bi,m with i ≤ n and m ∈ N then by Lemma 9

m−1

bn+1,m ≤ (N nk)2m−1 + N nk−1 (N nk)2p−1(N nk)2(m−p)−1

p=1

= N 2m−1n(2m−1)k + N nk−1(m − 1)N 2m−2n2(m−2)k

≤ N 2m−1(n(2m−1)k + (2m − 1)kn(2m−1)k−1) ≤ N 2m−1(n + 1)(2m−1)k

by recognizing the leading terms in the binomial expansion of (n + 1)(2m−1)k.

Remark 11. We will use the observation that these arguments establish these

same results also for the case where the coeﬃcients are bounded functions; in the

preceding arguments one merely replaces the coeﬃcients by their bounds.

4. Orbit estimates. With the notations of the proof of Theorem 4, we deﬁne

R+ := {(α1, . . . , αl, β1, . . . , βm) m βiµi − l αiλi = µs for some s ≤ m}.

i=1 i=1

Lemma 12. There exists p ∈ (0, 1/2) such that m βi ≥ p|τ | for any τ ∈ R+.

i=1

Proof. m βiµi − l αiλi = µs > 0 gives

i=1 i=1

ll m m

λ1 αi ≤ αiλi < βiµi ≤ µm βi

i=1 i=1 i=1 i=1

and l m m

i=1 i=1 i=1

|τ | = αi + βi < (1 + µm ) βi.

λ1

Thus, any p ≤ 1/(1 + µm ) is as required.

λ1

Lemma 13. There exist H, γ1 > 0 such that if δ is suﬃciently small and fj(x) <

δ for j = 0, . . . , n then |xi| < eH2 log δ−γ1n for each expanding coordinate xi.

Proof. Denote by (·)± the coordinate projections to the expanding and contracting

directions, respectively. There is a conjugacy h to f that is H-H¨older continuous

(Theorem 2) together with its inverse and preserves the contracting plane. Let

(x+, x−) = h((x+, x−)). If fj(x) < δ for j = 0, . . . , n then Lj(x ) < δH for

j = 0, . . . , n. For the linear part of f we have x+ < δH e−µ1n, so

x+ ≤ h((x+, x−)) − h((0, x−)) < δH2 e−γ1n

for some γ1 > 0.

Lemma 14. There exists γ > 0 such that if fj(x) < δ for j = 0, . . . , n then

cn,m(x) := max {|xτ | τ ∈ R+, |τ | = m} ≤ e−γ(n+Cδ)m x for m ≥ 2,

where Cδ → ∞ as δ → 0.

Proof. xτ has more than pm expanding components (counted with multiplicity) by

Lemma 12. Applying Lemma 13 to these and |xi| ≤ x to any one of the remaining

e−pm(γ1n−H2 log δ)

m−( pm + 1) > m − ( m + 1) ≥ m −1 ≥ 0 gives |xτ | < x .

2 2

Remark 15. By Lemma 10 there exists C > 0 such that bn,m ≤ eγ(n+C)m/2, and

we henceforth take δ > 0 such that Cδ > C.

DIFFERENTIABILITY OF THE HARTMAN–GROBMAN LINEARIZATION 983

5. The conjugacy. Start with the Bronstein–Kopanski˘ı conjugacy H+ to f from

Theorem 4. Consider a nonincreasing C∞ “bump” function ϕ : [0, 1] → [0, 1] with

ϕ(1/4) = 1 and ϕ(3/4) = 0, and multiply the quadratic terms of f by ϕ( x /δ).

Near 0 this new map ˜f is conjugate to f by the identity. If x ≥ δ then ˜f(x) = Lx.

Take δ < δ suﬃciently small, and for x < δ deﬁne

n+(x) := 1 + max{n ∈ N ˜fi(x) < δ for 0 ≤ i ≤ n}.

Then lim x →0 n+(x) = ∞ and n+(˜f(x)) = n+(x) − 1. Since the linear part of

f is hyperbolic, n+ is ﬁnite oﬀ the contracting direction. For x < δ and with

hn := L−n˜fn as in Remark 8 we set h+ = 0 on the contracting direction and

h+(x) := (hn+(H+(x))(H+(x)))+,

Continuity on the contracting direction follows from the last sentence in the proof

of Lemma 16. Discontinuities of n+ do not produce discontinuities of h+ because

hn+(H+(x))(H +(x)) = hn+(H+(x))+1(H +(x)).

Theorem 4 applied to f −1 yields a conjugacy to a quadratic polynomial. To this

our intermediate results also apply, and hence we can deﬁne n− and h− analogously

to n+ and h+ and set h := (h+, h−). The next two results for h+ combined with

the analogous ones for h− (which we omit) show that h is diﬀerentiable at 0 and is

a conjugacy between f and its linear part L.

Lemma 16. Dh+(0) = Id, i.e., (h+(x) − x)+ = o( x ).

Proof. Since H+(x) − x = o( x ) by Theorem 4, and o( H+(x) ) ⊂ o( x ) we

show (hn+(x)(x) − x)+ = o( x ). With cn,m from Lemma 14

(hn+(x)(x)−x)+ ≤ |αi,n+,τ |·|xτ | ≤ |αi,n+,τ |·cn+,m(x)

expanding τ ∈R+ m≥2 |τ |=m

coordinates i |τ |≥2 τ ∈R+

≤ cn+,m(x)bn+,m ≤ e−γ(n++Cδ)meγ(n++C)m/2 x

m≥2 m≥2

by Remark 11 and Lemma 10 with α = γ/2. Remark 15 and q := e−γ(n++C)/2 give

(hn+(x)(x)−x)+ ≤ m≥2 qm x = x q2/(1 − q). If x+ → 0 then n+(x) → ∞

and q → 0, so (hn+(x)(x) − x)+ = o( x ), hence (h(x) − x)+ = o( x ).

Analogously to our other notations we write L = L+ ⊕ L−. Then

(h(f (x)))+ = (L−n+(H+(f(x))) ˜fn+(H+(f(x)))(H+(f (x))))+

= (L−n+(˜f(H+(x))) ˜fn+(˜f(H+(x))) (˜f(H +(x))))+

= (L−n+(H+(x))+1 ˜fn+(H+(x)) (H +(x)))+

= L−+n+(H+(x))+1 (˜fn+(H+(x)) (H +(x)))+

= L+(L−n+(H+(x)) ˜fn+(H+(x)) (H+(x)))+ = L+h+(x) = (Lh(x))+.

This observation and its counterpart for h− give h ◦ f = L ◦ h for small x, i.e., h

conjugates f and L in a neighborhood of 0. Since Lemma 16 and its counterpart

for h− imply invertibility near 0, this completes the proof of Theorem 3.

984 MISHA GUYSINSKY, BORIS HASSELBLATT, AND VICTORIA RAYSKIN

REFERENCES

[B] Genrich Ruvimovich Belitski˘ı: On the Grobman–Hartman theorem in the class Cα, preprint

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tiﬁc Publishing Co., Inc., River Edge, NJ, 1994

[G] David M. Grobman: Homeomorphism of systems of diﬀerential equations, Doklady Akad.

Nauk SSSR 128 (1959) 880–881; Topological classiﬁcation of neighborhoods of a singularity

in n-space, Mat. Sbornik 56, no. 98, (1962), 77–94

[H1] Philip Hartman: On local homeomorphisms of Euclidean spaces, Bolet´ın de la Sociedad

Matem´atica Mexicana. (2) 5 (1960), 220–241

[H2] Philip Hartman: A lemma in the theory of structural stability of diﬀerential equations,

Proceedings of the American Mathematical Society 11 (1960), 610–620; On the local lin-

earization of diﬀerential equations Proceedings of the American Mathematical Society 14

(1963) 568–573

[KH] Anatole B. Katok, Boris Hasselblatt: Introduction to the modern theory of dynamical sys-

tems, Cambridge University Press, 1995

[S] Sebastian van Strien: Smooth linearization of hyperbolic ﬁxed points without resonance

conditions, Journal of Diﬀerential Equations 85 (1990), no. 1, 66–90

Received July 2002; revised November 2002.

Department of Mathematics, The Pennsylvania State University, University Park,

PA 16802-6401

E-mail address: guysin [email protected]

Department of Mathematics, Tufts University, Medford, MA 02155-5597

E-mail address: [email protected]

Department of Mathematics, 360 Portola Plaza, MS Building, University of Cali-

fornia, Los Angeles, CA 90095

E-mail address: [email protected]