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APPEARED IN BULLETIN OF THE

arXiv:math/9210220v1 [math.FA] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 217-238PREVALENCE: A TRANSLATION-INVARIANT “ALMOSTEVERY”ON INFINITE-DIMENSIONAL SPACESBRIAN R. HUNT, TIM SAUER, AND JAMES A. YORKEAbstract. We present a measure-theoretic condition for a property to hold“almost everywhere” on an infinite-dimensional vector space, with particularemphasis on function spaces such as Ck and Lp.

Like the concept of “Lebesguealmost every” on finite-dimensional spaces, our notion of “prevalence” is trans-lation invariant. Instead of using a specific measure on the entire space, wedefine prevalence in terms of the class of all probability measures with com-pact support.

Prevalence is a more appropriate condition than the topologicalconcepts of “open and dense” or “generic” when one desires a probabilisticresult on the likelihood of a given property on a function space. We give sev-eral examples of properties which hold “almost everywhere” in the sense ofprevalence.

For instance, we prove that almost every C1 map on Rn has theproperty that all of its periodic orbits are hyperbolic.1. IntroductionUnder what conditions should it be said that a given property on an infinite-dimensional vector space is virtually certain to hold?

For example, how are state-ments of the following type made mathematically precise? (1) Almost every function f: [0, 1] →R in L1 satisfiesR 10 f(x)dx ̸= 0.

(2) Almost every sequence {ai}∞i=1 in l2 has the property that P∞i=1 ai diverges. (3) Almost every C1 function f: R →R has the property that f ′(x) ̸= 0 wheneverf(x) = 0.

(4) Almost every continuous function f: [0, 1] →R is nowhere differentiable. (5) If A is a compact subset of Rn of box-counting dimension d, then for 1 ≤k ≤∞, almost every Ck function f: Rn →Rm is one-to-one on A, provided thatm > 2d.

(When A is a C1 manifold, the conclusion can be strengthened to say thatalmost every f is an embedding. )(6) If A is a compact subset of Rn of Hausdorffdimension d, then for 1 ≤k ≤∞, almost every Ck function f: Rn →Rm has the property that the Hausdorffdimension of f(A) is d, provided that m ≥d.Received by the editors Dec. 22, 1991, and in revised form, May 5, 1992.1991 Mathematics Subject Classification.

Primary 28C20, 60B11; Secondary 58F14.The first author was supported by the NSWC Independent Research Program. The second andthird authors were partially supported by the U.S. Department of Energy (Scientific ComputingStaff, Office of Energy Research).c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2B. R. HUNT, TIM SAUER, AND J.

A. YORKE(7) For 1 ≤k ≤∞, almost every Ck map on Rn has the property that all of itsfixed points are hyperbolic (and further, that its periodic points of all periods arehyperbolic). (8) For 4 ≤k ≤∞, almost every Ck one-parameter family of vector fields on R2has the property that as the parameter is varied, every Andronov-H¨opf bifurcationwhich occurs is “typical” (in a sense to be made precise later).In Rn, there is a generally accepted definition of “almost every”, which is thatthe set of exceptions has Lebesgue measure zero.

The above statements requirea notion of “almost every” in infinite-dimensional spaces. We will be concernedmainly with function spaces such as Lp for 1 ≤p ≤∞and Ck for (integers)0 ≤k ≤∞on subsets of Rn; many of these spaces are Banach spaces, and all havea complete metric.

The following are some properties of “Lebesgue measure zero”and “Lebesgue almost every” which we would like to preserve on these spaces. (i) A measure zero set has no interior (“almost every” implies dense).

(ii) Every subset of a measure zero set also has measure zero. (iii) A countable union of measure zero sets also has measure zero.

(iv) Every translate of a measure zero set also has measure zero.One could define “almost every” on a given function space in terms of a specificmeasure. For example, the Wiener measure on the continuous functions is appro-priate for some problems.

However, the notion of “almost every” with respect tosuch a measure violates property (iv). The following paragraph illustrates some ofthe difficulties involved in defining an analogue of Lebesgue measure on functionspaces.

We assume all measures to be defined (at least) on the Borel sets of thespace.In an infinite-dimensional, separable1 Banach space, every translation-invariantmeasure which is not identically zero has the property that all open sets haveinfinite measure. To see this, suppose that for some ε, the open ball of radius εhas finite measure.

Because the space is infinite dimensional, one can construct aninfinite sequence of disjoint open balls of radius ε/4 which are contained in the ε-ball. Each of these balls has the same measure, and since the sum of their measuresis finite, the ε/4-balls must have measure 0.

Since the space is separable, it canbe covered with a countable collection of ε/4-balls, and thus the whole space musthave measure 0. (Even if the space were not separable, we would be left with theundesirable property that some open sets have measure zero, violating property (i)above.

)In the absence of a reasonable translation-invariant measure on a given functionspace, one might hope there is a measure which at least satisfies condition (iv)above; such a measure is called quasi-invariant. In Rn, there are an abundance offinite measures which are quasi-invariant, such as Gaussian measure.

However, foran infinite-dimensional, locally convex topological vector space, a σ-finite,2 quasi-invariant measure defined on the Borel sets must be identically zero [5, 31, 32] (seealso pp. 138–143 of [35]).1By separable we mean that the space has a countable dense subset.2By σ-finite we mean that the entire space can be expressed as a countable union of sets offinite measure.

This rules out measures such as “counting measure”, which assigns to each set itscardinality.

PREVALENCE: A TRANSLATION-INVARIANT3Rather than search for a partial analogue of Lebesgue measure on functionspaces, our strategy is to find an alternate characterization of the concepts of“Lebesgue measure zero” and “Lebesgue almost every” which has a natural exten-sion to function spaces. Properties (i)–(iv) alone do not uniquely determine theseconcepts, but there is a more subtle property which does.

In Rn, let us considerthe class of “probability measures with compact support”, that is, those measuresµ for which there exists a compact set T ⊂Rn such that µ(T ) = µ(Rn) = 1. (v) Let S be a Borel set.

If there exists a probability measure µ with compactsupport such that every translate of S has µ-measure zero, then S hasLebesgue measure zero.Property (v) is proved in §2 (see Fact 6) by a simple application of the Tonellitheorem (a variant of the Fubini theorem [4]). Notice that conversely, if S ⊂Rnhas Lebesgue measure zero, then the hypothesis of property (v) is satisfied with µequal to (for instance) the uniform probability measure on the unit ball.Given a probability measure µ with compact support, we can define a trans-lation-invariant measure ˜µ on Borel sets S by ˜µ(S) = 0 if every translate of S hasµ-measure zero and ˜µ(S) = ∞otherwise.

What property (v) above says is thatevery such measure ˜µ is greater than or equal to Lebesgue measure on the Borel setsof Rn. Thus one way to show that a Borel set is small, in a translation-invariantprobabilistic sense, is to show that ˜µ(S) = 0 for some µ.

Such a strategy is plausibleon infinite-dimensional spaces because it is not hard to find probability measureswith compact support (for example, uniform measure on a line segment, or on theunit ball of any finite-dimensional subspace).In general, we will call a Borel set “shy” if ˜µ(S) = 0 for some probability measureµ with compact support, and we call any other set shy if it is contained in a shy Borelset (just as every Lebesgue measure zero set is contained in a Lebesgue measurezero Borel set). We then define a “prevalent” set to be a set whose complement isshy.

This definition may not characterize all sets for which the label “almost every”is appropriate; our claim is rather that properties which hold on prevalent sets areaccurately described as holding “almost everywhere”.In the absence of a probabilistic notion of “almost every”, statements such as 1–8above have often been formulated in terms of the topological notion of “genericity”.In this terminology, a property on a complete metric space is said to be generic if theset on which it holds is residual, meaning that it contains a countable intersection ofopen dense sets.3 The complement of a residual set is said to be of the first category;equivalently, a first category set is a countable union of nowhere dense sets. Thenotion of “first category” was introduced by Baire in 1899, and his category theoremensures that a residual subset of a complete metric space is nonempty, and in factdense [20].The concepts of “first category” and “generic” have formal similarities to “mea-sure zero” and “almost every”, satisfying a set of properties analogous to (i)–(iv)above.

They also agree for some sets in Rn; for example, the set of rational numbershas measure zero and is of the first category. But perhaps too much emphasis hasbeen placed on those examples in which first category sets happen to have measure3Many authors require a residual set to be (not just contain) a countable intersection of opendense sets.

Our terminology follows [20].

4B. R. HUNT, TIM SAUER, AND J.

A. YORKEzero. Sets which are open and dense in Rn can have arbitrarily small Lebesguemeasure, and residual sets can have measure zero.In fact, many properties are known to be topologically generic in Rn but havelow probability.

While the reader may be able to provide examples from his or herown experience, we include some for completeness.Example 1. For n ≥1 let Un = {x ∈[0, 1] : 0 < 2nx (mod 1) < 2−n}.

Noticethat Vm = Sn>m Un is open and dense but has measure less than 2−m. Hencegenerically points in [0, 1] satisfy 0 < 2nx (mod 1) < 2−n for infinitely many valuesof n, but the set of such points (Tm≥1 Vm) has measure zero.

A similar constructionarises naturally in [11].Example 2. Here we consider how well real numbers can be approximated byrationals.

The Liouville numbers are the real numbers λ which have the propertythat for all c, n > 0 there exist integers p and q > 0 such thatλ −pq < cqn .As in the previous example, the set of Liouville numbers is residual but has Lebesguemeasure zero [20]. In contrast are the Diophantine numbers, real numbers µ whichhave the property that for every ε > 0 there exists a c > 0 such that for all integersp and q > 0,µ −pq >cq2+ε .The set of Diophantine numbers is of the first category but has full Lebesgue mea-sure in every interval.Example 3.

Arnold studied the family of diffeomorphisms on a circlefω,ε(x) = x + ω + ε sin x(mod 2π),where 0 ≤ω ≤2π and 0 ≤ε < 1 are parameters. For each ε we can define the setSε = {ω ∈[0, 2π] : fω,ε has a stable periodic orbit}.For 0 < ε < 1, the set Sε is a countable union of disjoint open intervals (one foreach rational rotation number), and is an open dense subset of [0, 2π].

However,the Lebesgue measure of Sε approaches zero as ε →0. For small ε, the probabilityof picking an ω in this open dense set is very small.

See pp. 108–109 of [1] for moredetails.Example 4.

Consider the dynamics of an analytic map in the complex plane neara neutral fixed point. Suppose the fixed point is the origin; then the map can bewritten in the formz 7→e2πiαz + z2f(z)with 0 ≤α ≤1 and f(z) analytic.

Siegel [30] proved that for Lebesgue almostevery α (specifically, if α is not a Liouville number), the above map is conjugate toa rotation in a neighborhood of the origin under an analytic change of coordinates.On the other hand, Cremer [3] previously showed that if f is a polynomial (notidentically zero), then for a residual set of α the above map is not conjugate to

PREVALENCE: A TRANSLATION-INVARIANT5a rotation in any neighborhood of the origin. These results are discussed on pp.98–105 of [2].Example 5.

Consider the map z 7→ez on the complex plane. Misiurewicz [16]proved that this map is “topologically transitive”, which implies that a residualset of initial conditions have dense trajectories.

On the other hand, Lyubich [13]and Rees [24] showed that Lebesgue almost every initial condition has a trajectorywhose limit set lies on the real axis (in fact, the limit set is just the trajectory of0). See [14] for a discussion of both results.Example 6.

For many families of dynamical systems in R2 depending on a param-eter, Newhouse [18] and Robinson [25] constructed a set of parameters for whichinfinitely many attractors coexist. The constructed set is residual in an interval,but is shown in [33] and [19] to have measure zero.In view of these examples, one might ask why the concept of “residual” is used.Sometimes, one just wants to show that a set obtained by a countable intersectionis nonempty, or further that it is dense.

For example, the existence of continuousbut nowhere differentiable functions can be proved by showing that they form aresidual subset of the continuous functions; this argument is due to Banach (see§III.34.VIII of [12]). Other times, one wants to show that a set is “large” in atopological sense, perhaps because there has been no probabilistic alternative.

Theconcept of “prevalence” is intended for situations where a probabilistic result isdesired.In §2 we formally define prevalence, shyness (the opposite of prevalence), andrelated concepts, and develop some of the basic theory of these notions. Section 3examines the eight statements from the beginning of this section in the new frame-work.

In §4 we develop some of the theory of “transversality” (between functionsand manifolds) in the context of prevalence, and use it to prove the third, seventh,and eighth statements. Finally, §5 discusses some other ideas related to prevalence.2.

PrevalenceLet V be a complete metric linear space, by which we mean a vector space (realor complex) with a complete metric for which addition and scalar multiplication arecontinuous. When we speak of a measure on V we will always mean a nonnegativemeasure that is defined on the Borel sets of V and is not identically zero.

We writeS + v for the translate of a set S ⊂V by a vector v ∈V .Definition 1. A measure µ is said to be transverse to a Borel set S ⊂V if thefollowing two conditions hold:(i) There exists a compact set U ⊂V for which 0 < µ(U) < ∞.

(ii) µ(S + v) = 0 for every v ∈V .Condition (i) ensures that a transverse measure can always be restricted to afinite measure on a compact set (see Fact 2 below), and in developing the theory oftransverse measures it is often useful to think in terms of probability measures withcompact support. For applications it will be convenient to use measures which (likeLebesgue measure) are neither finite nor have compact support.

If V is separable,then all measures which take on a value other than 0 and ∞can be shown to satisfycondition (i) [21].

6B. R. HUNT, TIM SAUER, AND J.

A. YORKEDefinition 2. A Borel set S ⊂V is called shy4 if there exists a measure transverseto S. More generally, a subset of V is called shy if it is contained in a shy Borelset.

The complement of a shy set is called a prevalent set.Strictly speaking, the above concepts could be called “translation shy” and“translation prevalent”. On manifolds for which there is no distinguished set oftranslations, the corresponding theory is more difficult; this is a topic we do notaddress in this paper.

We again emphasize that the definitions of shy and prevalentwould be unchanged if we required transverse measures to be probability measureswith compact support.Roughly speaking, the less concentrated a measure is, the more sets it is trans-verse to. For instance, a point mass is transverse to only the empty set.

Also,we will later show (see Fact 6) that if any measure is transverse to a Borel setS ⊂Rn, then Lebesgue measure is transverse to S too. When V is infinite di-mensional, a convenient choice for a transverse measure is often Lebesgue measuresupported on a finite-dimensional subspace.5 For example, Lebesgue measure onthe one-dimensional space spanned by a vector w ∈V is transverse to a Borel setS ⊂V if for all v ∈V , the set of λ ∈R (or C if V is complex) for which v + λw ∈Shas Lebesgue measure zero.

It immediately follows that every countable set in Vis shy, and every proper subspace of V is shy. Notice that because it is possible torepresent an infinite-dimensional space as the continuous linear image of a propersubspace, the continuous linear image of a shy set need not be shy.We now present some important facts about transversality and shyness.

Thefirst follows immediately from the above definitions, and in particular implies thatprevalence is translation invariant.Fact 1. If S is shy, then so is every subset of S and every translate of S.Fact 2.

Every shy Borel set S has a transverse measure which is finite with compactsupport. Furthermore, the support of this measure can be taken to have arbitrarilysmall diameter.Proof.

Let µ be a measure transverse to a Borel set S ⊂V . Then by condition (i)of Definition 1 it can be restricted to a compact set U of finite and positive measure,and the restriction is certainly also transverse to S. Also, since U is compact it canbe covered for each ε > 0 by a finite number of balls of radius ε, and at least oneof these balls must intersect U in a set of positive measure.

The intersection of Uwith the closure of this ball is compact, and the restriction of µ to this set is alsotransverse to S.An immediate consequence of Fact 2 is that a shy Borel set has no interior. Thesame is then true of every shy set, since every shy set is contained in a shy Borelset.

Hence we have the following fact.Fact 2′. All prevalent sets are dense.4The word “shy” was suggested to us by J. Milnor.5An exact characterization of Lebesgue measure on a given finite-dimensional subspace dependson the choice of a basis for the subspace, but since we are only interested in whether or not setshave measure zero, the choice of basis is unimportant for our purposes.

PREVALENCE: A TRANSLATION-INVARIANT7Next, we would like to know that the union of two shy sets is also shy. GivenBorel sets S, T ⊂V containing the original sets and measures µ and ν transverseto S and T respectively, we must then find a measure which is transverse to bothS and T .

We can assume by Fact 2 that µ and ν are finite with compact support.Then the measure we desire is the convolution µ ∗ν of µ and ν, defined as follows.Definition 3. Let µ and ν be measures on V .

Let µ × ν be the product measureof µ and ν on the Cartesian product V × V , and for a given Borel set S ⊂V letSΣ = {(x, y) ∈V × V : x + y ∈S}. Then SΣ is a Borel subset of V × V , and wedefine µ ∗ν(S) = µ × ν(SΣ).If µ and ν are finite, then µ × ν is finite, and the characteristic function of SΣ isintegrable with respect to µ × ν.

Then by the Fubini theorem [4],µ ∗ν(S) =ZVµ(S −y) dν(y) =ZVν(S −x) dµ(x).We thus have the following fact.Fact 3. Let µ and ν be finite measures with compact support.

If µ is transverse toa Borel set S, then so is µ ∗ν.6From Fact 3 it follows that the union of two shy sets is shy, and more generallythe following fact holds.Fact 3′. The union of a finite collection of shy sets is shy.Fact 3′ extends to countable unions by a slightly more complicated argument.Fact 3′′.

The union of a countable collection of shy sets is shy.Proof. Given a countable collection of shy subsets of V , let S1, S2, .

. .

be shy Borelsets containing the original sets. Let µ1, µ2, .

. .

be transverse to S1, S2, . .

. , respec-tively.

By Fact 2, we can assume without loss of generality that each µn is finiteand supported on a compact set Un with diameter at most 2−n. By normalizingand translating the measures, we can also assume that µn(V ) = 1 for all n andthat each Un contains the origin.

With these assumptions we can define a measureµ which is essentially the infinite convolution of the µn. We rely on the theory ofinfinite product measures; see pp.

200–206 of [4] for details.The infinite Cartesian product U Π = U1 × U2 × · · · is compact by the Tychonofftheorem [4] and has a product measure µΠ = µ1 × µ2 × · · · defined on its Borelsubsets, with µΠ(U Π) = 1. Since V is complete and each vector in Un lies at most2−n away from zero, there is a continuous mapping from U Π into V defined by(v1, v2, .

. . ) 7→v1 + v2 + · · · .

The image U of U Π under this mapping is compact,and µΠ induces a measure µ supported on U, given by µ(S) = µΠ(SΣ), whereSΣ = {(v1, v2, . .

. ) ∈U Π : v1 + v2 + · · · ∈S}.

We will be done if we show that µ istransverse to every Sn.Since the Cartesian product of measures is associative (and commutative), wecan write µΠ = µn × νΠn with νΠn = µ1 × · · · × µn−1 × µn+1 × · · · . Let νn be themeasure on V induced by νΠn under the summation mapping (as µ was induced6Notice that µ ∗ν has compact support because its support is contained in the continuousimage, under the mapping (x, y) 7→x + y, of the Cartesian product of the supports of µ and ν.

8B. R. HUNT, TIM SAUER, AND J.

A. YORKEby µΠ). Then µ = µn ∗νn, and therefore by Fact 3, µ is transverse to Sn.

Thiscompletes the proof.We are now in a position to show that the conditions for shyness given in the be-ginning of this section can be weakened in some cases. First, consider the followingdefinition.Definition 4.

A measure µ is essentially transverse to a Borel set S ⊂V if condi-tion (i) of Definition 1 holds and µ(S + v) = 0 for a prevalent set of v ∈V .Though essential transversality is weaker than transversality, the following factholds.Fact 4. If a Borel set S ⊂V has an essentially transverse measure, then S is shy.Proof.

Let µ be a measure that is essentially transverse to S. As in Fact 2 we mayassume µ is finite with compact support. The set of all v ∈V for which µ(S−v) > 0is shy, and hence is contained in a shy Borel set T .

Let ν be a finite measure withcompact support which is transverse to T . Then for all w ∈V ,µ ∗ν(S + w) =ZVµ(S + w −y) dν(y) = 0since the integrand is nonzero only on a subset of T + w and ν(T + w) = 0.

Thusµ ∗ν is transverse to S, and S is shy.Next let us examine a local definition of shyness and prevalence.Definition 5. A set S ⊂V is locally shy if every point in the space V has aneighborhood whose intersection with S is shy.A set is locally prevalent if itscomplement is locally shy.Facts 1, 2′, and 3′ immediately hold also for local shyness and local prevalence,but whether Fact 3′′ does is not clear in general.

If V is separable though, it turnsout that the local definitions of shyness and prevalence are equivalent to the globaldefinitions. (On the other hand, it is not clear that these notions are the same inspaces such as L∞and l∞.

)Fact 5. All shy sets are locally shy.

If V is separable, all locally shy subsets of Vare shy.Proof. The first part of this fact is trivial.

To verify the second part, recall thatby the Lindel¨of theorem [4], if V is a separable metric space then every open coverof V has a countable subcover. Given a locally shy set S ⊂V , the neighborhoodswhose intersections with S are shy cover V .

Hence by taking a countable subcover,S can be written as a countable union of shy sets. Thus by Fact 3′′, S is shy.If V is finite dimensional, then shyness and local shyness are equivalent by Fact5.

In this case we can show further that both of these concepts are equivalent tohaving Lebesgue measure zero.Fact 6. A set S ⊂Rn is shy if and only if it has Lebesgue measure zero.

PREVALENCE: A TRANSLATION-INVARIANT9Proof. We need only consider Borel sets, because every Lebesgue measure zero setis contained in a Borel set with measure zero.

If a Borel set S has Lebesgue measurezero, then Lebesgue measure is transverse to S, and S is shy. On the other hand, ifa Borel set S is shy, then by Fact 2 there is a finite measure µ which is transverseto S. Let ν be Lebesgue measure.

Though ν is not finite, it is σ-finite, so by theTonelli theorem [4] we have (as in the equation preceding Fact 3) that0 =ZRn µ(S −y) dν(y) =ZRn ν(S −x) dµ(x) = µ(Rn)ν(S).In other words, S has Lebesgue measure zero.Fact 6 implies that in Rn, Lebesgue measure is a best possible candidate tobe transverse to a given Borel set. As we mentioned earlier, when looking for atransverse measure in an infinite-dimensional space, a useful type of measure to tryis Lebesgue measure supported on some finite-dimensional subspace.Definition 6.

We call a finite-dimensional subspace P ⊂V a probe for a set T ⊂Vif Lebesgue measure supported on P is transverse to a Borel set which contains thecomplement of T .Then a sufficient (but not necessary) condition for T to be prevalent is for it tohave a probe. One advantage of using probes is that a single probe can often beused to show that a given property is prevalent on many different function spacesby applying the following simple fact.Fact 7.

If µ is transverse to S ⊂V and the support of µ is contained in a subspaceW ⊂V , then S ∩W is a shy subset of W.Next we use one-dimensional probes to show that all compact subsets of aninfinite-dimensional space are shy. We prove in fact that given a compact set S ⊂V ,there are one-dimensional subspaces L for which every translate of L intersects Sin at most one point.

To do this we show that a residual set of vectors in V spanone-dimensional subspaces L with the above property. Here then is an applicationof the fact that a residual set is nonempty.Fact 8.

If V is infinite dimensional, all compact subsets of V are shy.Proof. We assume V is a real vector space; the proof is nearly identical for a complexvector space.

Let S ⊂V be a compact set, and define the function f : R×S×S →Vbyf(α, x, y) = α(x −y).If a vector v ∈V is not in the range of f, then v spans a line L such that everytranslate of L meets S in at most one point; in particular, L is a probe for thecomplement of S. We then need only show that the range of f is not all of V ;we show in fact that it is a first category set. For each positive integer N, the set[−N, N] × S × S is compact, and hence so is its image under f. Thus the range off is a countable union of compact sets.

Since V is infinite dimensional, a compactset in V has no interior (see p. 23 of [29]), and is then nowhere dense (because it isclosed). Therefore the range of f is of first category as claimed.

10B. R. HUNT, TIM SAUER, AND J.

A. YORKE3. Applications of prevalenceFrom now on, when we say “almost every” element of V satisfies a given property,we mean that the subset of V on which the property holds is prevalent.

Given thisterminology, the eight numbered statements from the introduction can be provedby constructing appropriate probes (see Definition 6).Proposition 1. Almost every function f : [0, 1] →R in L1 satisfies R 10 f(x) dx ̸=0.A probe for Proposition 1 is the one-dimensional space of all constant functions.Notice that this probe is contained in Ck for 0 ≤k ≤∞, so the above propertyalso holds for almost every f in Ck.

Similar remarks can be made about most ofthe results below.Proposition 2. For 1 < p ≤∞, almost every sequence {ai}∞i=1 in lp has theproperty that P∞i=1 ai diverges.For Proposition 2, the one-dimensional space spanned by the element {1/i}∞i=1 ∈lp is a probe for each 1 < p ≤∞.The third statement in the introduction can be proved using the space of constantfunctions as a probe; this follows immediately from the Sard theorem [26].

Here westate a more general result, which uses a higher-dimensional probe. We write f (i)for the ith derivative of f.Proposition 3.

Let k be a positive integer. Almost every Ck function f: R →Rhas the property that at each x ∈R, at most one of {f (i)(x) : 0 ≤i ≤k} is zero.The space of polynomials of degree ≤k is a probe for Proposition 3, as we willprove in the next section.

By Fact 3′′, Proposition 3 has the following corollary.Proposition 3′. Almost every C∞function f: R →R has the property that ateach x ∈R, at most one of {f (i)(x) : i ≥0} is zero.Because the dimension of the probe used to prove Proposition 3 goes to infinityas k →∞, it is not clear whether Proposition 3′ can be proved directly using aprobe.Proposition 4.

Almost every continuous function f: [0, 1] →R is nowhere differ-entiable.Proposition 4 requires a two-dimensional probe. A one-dimensional probe wouldbe spanned by a continuous function g with the property that for all continuousf: [0, 1] →R, the function f + λg is nowhere differentiable for almost every λ ∈R.But if f(x) = −xg(x), then f + λg is differentiable at x = λ for every λ between0 and 1.

The proof of Proposition 4 uses a probe spanned by a pair of functions gand h for which λg + µh is nowhere differentiable for every (λ, µ) ∈R2 aside fromthe origin [9].Next we state a prevalence version of the Whitney Embedding Theorem.Proposition 5. Let A be a compact C1 manifold of dimension d contained in Rn.For 1 ≤k ≤∞, almost every Ck function f: Rn →R2d+1 is an embedding of A.

PREVALENCE: A TRANSLATION-INVARIANT11The probe used in the proof of Proposition 5 is the space of linear functions fromRn to R2d+1. Whitney [34] showed that a residual subset of the Ck functions fromRn to R2d+1 are embeddings of A.

This result was preceded by a topological versiondue to Menger in 1926 (see p. 56 of [10]), which states that for a compact space Aof topological dimension d, a residual subset of the continuous functions from A toR2d+1 are one-to-one. Proposition 5, and the following generalization to compactsubsets of Rn which need not be manifolds (or even have integer dimension), areproved in [28].Proposition 5′.

If A is a compact subset of Rn of box-counting (capacity) dimen-sion d, and 1 ≤k ≤∞, then almost every Ck function f: Rn →Rm is one-to-oneon A, provided that m > 2d.Our next proposition concerns the preservation of Hausdorffdimension undersmooth transformations. Once again the probe is the space of all linear functionsfrom Rn to Rm; see [27] for a proof.Proposition 6.

If A is a compact subset of Rn of Hausdorffdimension d, and 1 ≤k ≤∞, then for almost every Ck function f: Rn →Rm the Hausdorffdimensionof f(A) is d, provided that m ≥d.Remark. It is interesting that Proposition 5′ fails for Hausdorffdimension (see [28]),and Proposition 6 fails for box-counting dimension (see [27]), under any reasonabledefinition of “almost every”.We now present a result about the prevalence of hyperbolicity for periodic orbitsof maps.

We say that a period p point of a map f: Rn →Rn is hyperbolic if thederivative of the pth iterate of F at the point has no eigenvalues (real or complex)with absolute value 1.Proposition 7. Let p be a positive integer.

For 1 ≤k ≤∞, almost every Ck mapon Rn has the property that all of its periodic points of period p are hyperbolic.Proposition 7 is proved in the next section using the space of polynomial func-tions of degree at most 2p −1 as a probe. Proposition 7 and Fact 3′′ imply thefollowing more elegant result.Proposition 7′.

For 1 ≤k ≤∞, almost every Ck map on Rn has the propertythat all of its periodic points are hyperbolic.Next consider one-parameter families of dynamical systems. As the parametervaries, it is likely that nonhyperbolic points will be encountered, and at such pointsbifurcations (creation or destruction of periodic orbits, or changes in stability oforbits) can occur.In general one can expect to prove results of the sort thatfor dynamical systems of a given type, almost every one-parameter family hasthe property that all of its bifurcations are “nondegenerate” in some fashion.

Acomplete discussion of such results is beyond the scope of this paper, but we includeas an example a result about Andronov-H¨opf bifurcations for flows in the plane.For flows (as opposed to maps), a fixed point is hyperbolic if the linear part of thevector field at the fixed point has no eigenvalues on the imaginary axis. Generally,a zero eigenvalue results in a saddle-node bifurcation and a pair of nonzero, pureimaginary eigenvalues results in an Andronov-H¨opf bifurcation; see [6] for details.The following proposition is proved in the next section.

12B. R. HUNT, TIM SAUER, AND J.

A. YORKEProposition 8. For 4 ≤k ≤∞, almost every Ck one-parameter family of vec-tor fields f(µ, x): R × R2 →R2 has the property that whenever f(µ0, x0) = 0 andDxf(µ0, x0) has nonzero, pure imaginary eigenvalues, the flow ˙x = f(µ, x) un-dergoes a nondegenerate Andronov-H¨opf bifurcation in the sense that the followingconditions hold in a neighborhood U of (µ0, x0):(i) The fixed points in U form a curve (µ, x(µ)), where x(µ) is Ck.

(ii) The point (µ, x(µ)) is attracting when µ is on one side of µ0 and repellingwhen µ is on the other side. (iii) There is a Ck−2 surface7 of periodic orbits in R×R2 which has a quadratictangency with the plane µ = µ0.

The periodic orbits are attracting if thefixed points for the same parameter values are repelling, and are repellingif the corresponding fixed points are attracting.4. Transversality and prevalenceThe proofs of Propositions 3, 7, and 8 are based on the idea of “transversality”,which we will discuss now in the context of functions from one Euclidean space toanother.

Given 1 ≤k ≤∞and 0 ≤d < ∞, we call M ⊂Rn a Ck manifold ofdimension d if for all x ∈M there is an open neighborhood U ⊂Rn of x and a Ckdiffeomorphism ϕ: U →V ⊂Rn such that ϕ(M ∩U) = (Rd ×{0})∩V . The tangentspace to M at x, denoted by TxM, is defined to be the inverse image of Rd × {0}under Dϕ(x).

Notice that an open set in Rn is a C∞manifold of dimension n, withtangent space Rn at every point.Definition 7. Let A ⊂Rn and Z ⊂Rm be manifolds.

We say that a C1 functionF: A →Rm is transversal to Z if whenever F(x) ∈Z, the spaces DF(x)(TxA) andTF (x)Z span Rm.Remark. If DF(x) maps TxA onto Rm for all x ∈A, then F is transversal to everymanifold in Rm; in this case we say that F is a submersion.In our applications A is always an open set in Rn, so the results below arestated only for this case, though they remain valid for functions whose domains aresufficiently smooth manifolds.

A basic result is the following (see [7] for a proof).Theorem 1 (Parametric Transversality Theorem). Let B ⊂Rq and A ⊂Rn beopen sets.

Let F: B × A →Rm be Ck, and let Z be a Ck manifold of dimensiond in Rm, where k > max(n + d −m, 0). If F is transversal to Z, then for almostevery λ ∈B, the function F(λ, ·) : A →Rm is transversal to Z.Notice that if F: A →Rm is transversal to Z ⊂Rm and the codimension of Z(that is, m minus the dimension of Z) is greater than the dimension of A, thenF(A) cannot intersect Z.

This observation is the basis for the following generalscheme for proving results like Propositions 3, 7, and 8. To show that almost everyf in a space such as Ck(Rn) has a given property, construct a function F consistingof the derivatives of f up to a certain order, and let Z be a manifold defined bya set of n + 1 conditions which F must satisfy at some point in Rn in order for fnot to have the desired property.

By an appropriate generalization of Theorem 1,7The surface is proved to be Ck−2 in [15]. However, we suspect that this surface can actuallybe shown to be Ck−1, in which case this proposition applies to C3 vector fields also.

PREVALENCE: A TRANSLATION-INVARIANT13it will follow that for almost every f, there is no point in Rn at which F satisfiesthe undesirable conditions.Let us formalize the above procedure.Definition 8. Let A ⊂Rn be open, and let f: A →Rm be Cl.

For k ≤l, wedefine the k-jet of f at x, denoted jkf(x), to be the ordered pair consisting of xand the degree k Taylor polynomial of f at x. Then jkf is a Cl−k function fromA to a space Jk(Rn, Rm) = Rn × P k(Rn, Rm), where P k(Rn, Rm) is the space ofpolynomials of degree ≤k from Rn to Rm.

We writejkf(x) = (x, f(x), Df(x), . .

. , Dkf(x)),where the coordinates (f(x), Df(x), .

. .

, Dkf(x)) represent the (unique) polyno-mial in P k(Rn, Rm) with the same derivatives up to order k as f at x.Remark. We will later write Jk(Rn, Rm) = Jk−1(Rn, Rm) × bP k(Rn, Rm), wherebP k(Rn, Rm) can be thought of as the space of homogeneous degree k polynomialsfrom Rn to Rm.

More precisely, jkf(x) can be decomposed into (jk−1f(x), Dkf(x)),where Dkf(x) represents a degree k polynomial which is homogeneous in a coordi-nate system based at x.The following is an example of the type of result we need; it is a prevalenceversion of a result previously formulated in terms of genericity [8].Theorem 1′ (Jet Transversality Theorem). Let A ⊂Rn be open and let Z be aCr manifold in Jk(Rn, Rm) with codimension c, where r > max(n −c, 0).Fork + max(n −c, 0) < l ≤∞, almost every Cl function f : A →Rm has the propertythat jkf is transversal to Z.Proof.

Let P = P k(Rn, Rm), thinking of P for now as a subspace of the Cl functionsfrom A to Rm. We claim that P is a probe (see Definition 6) for the above property.For p ∈P, let fp(x) = f(x)+p(x), and define the function F: P ×A →Jk(Rn, Rm)by F(p, x) = jkfp(x).

Notice that F is a submersion, because the first n coordinatesof F are just x, and for a given x the remaining coordinates of F act as a translation(by the Taylor polynomial of f at x) on P. In particular, F is transversal to Z. Thenby Theorem 1, for almost every p ∈P, the function F(p, ·) = jkfp is transversal toZ, and therefore P is a probe as claimed.A special case of Theorem 1′ is the following prevalence version of the ThomTransversality Theorem.Corollary 1′′. Let A ⊂Rn be open and let Z be a Cr manifold in Rm with codi-mension c, where r > max(n −c, 0).

For max(n −c, 0) < k ≤∞, almost every Ckfunction f: A →Rm is transversal to Z.Propositions 3, 7, and 8 can be proved using Theorems 1 and 1′, except that wewould then have to assume in Proposition 3 that f is Ck+1 and in Proposition 7that the map is C2. Instead we use the following results, which do not require thoseadditional assumptions and also allow us to avoid determining the entire manifoldstructure of Z.

14B. R. HUNT, TIM SAUER, AND J.

A. YORKEDefinition 9. We say that a set S is a zero set in a manifold M of dimension d ifS ⊂M and for every x ∈M there is a neighborhood U of x and a diffeomorphism ϕon U which takes M ∩U to an open set in Rd and for which ϕ(S ∩U) has Lebesguemeasure zero in Rd.Remark.

Since sets of Lebesgue measure zero are preserved under diffeomorphism,the particular choice of ϕ in Definition 9 does not matter; that is, a zero set in Mhas Lebesgue measure zero with respect to all local C1 coordinate systems on M.We will need a Fubini-like result for zero sets of manifolds which allows us toprove that a Borel set is a zero set in M by showing that it is a zero set on theleaves of an appropriate foliation of M. See [22] for a general result of this type;for our purposes we need only the following simple lemma, which follows directlyfrom the Fubini theorem.Lemma 2. Let M be a manifold of dimension d, and let {Mα} be a partition ofM into manifolds of dimension d′ < d with the following property : every x ∈Mhas a neighborhood U and a diffeomorphism ϕ on U which maps M ∩U to an openset in Rd and which maps those Mα which intersect U into parallel hyperplanes ofdimension d′.

If Z is a Borel set in M and Z ∩Mα is a zero set in Mα for everyα, then Z is a zero set in M.Remark. The hypotheses of Lemma 2 are satisfied if M can be written as M1 ×M2with M1, M2 manifolds, and the partition of M consists of all manifolds of the form{x} × M2 with x ∈M1; this will usually be the case when we apply Lemma 2.We now present measure-theoretic analogues to Theorems 1 and 1′.Lemma 3 (Measure Transversality Lemma).

Let B ⊂Rq and A ⊂Rn be opensets, with points in B denoted by λ and points in A denoted by x. Let F: B ×A →Rm × Rs be a continuous function with components G: B × A →Rm andH: B × A →Rs.

Assume that the derivatives DλG, DxG, and DλH exist and arecontinuous at every point of B × A (but DxH need not exist). Let M be a manifoldin Rm with codimension n, and assume that for all x ∈A and all y ∈Rs, thefunction F(·, x) is transversal to M × {y}.

Let Z be a zero set in M × Rs. Thenfor almost every λ ∈B, there is no x ∈A for which F(λ, x) ∈Z.Remark.

The transversality hypothesis of Lemma 3 is automatically satisfied ifDλF has full rank at every point of B × A (that is, if F(·, x) is a submersion forevery x ∈A).Lemma 3 will be proved at the end of this section. Notice that in the case thatF is C1 and Z is a manifold with codimension greater than n, Lemma 3 is a specialcase of Theorem 1.

In much the same way as Theorem 1′ followed from Theorem1, the next theorem follows from Lemma 3.Theorem 3′ (Measure Jet Transversality Theorem). Assume k ≥1.

Let A ⊂Rnbe open and let M be a manifold in Jk−1(Rn, Rm) with codimension n. Let π be theprojection from Jk−1(Rn, Rm) onto its first n coordinates, and assume that π|M isa submersion. Let Z be a zero set in M × bP k(Rn, Rm) (see the remark followingDefinition 8).

Then for k ≤l ≤∞, almost every Cl function f : A →Rm has theproperty that the image of A under jkf does not intersect Z.

PREVALENCE: A TRANSLATION-INVARIANT15Remark. In our applications, M will be defined by a set of conditions that f and itsderivatives must satisfy at some point x.

When these conditions do not explicitlydepend on x, the hypothesis that π|M be a submersion is trivially satisfied.Proof. The proof is the same as for Theorem 1′, except that we must verify thatF(p, x) = jk(f(x) + p(x)), where p ∈P k(Rn, Rm), satisfies the transversality con-dition of Lemma 3.

Given y ∈bP k(Rn, Rm), we have by hypothesis that under pro-jection onto the first n coordinates in Jk(Rn, Rm), the tangent space to M × {y}at any point projects onto all of Rn. The remaining coordinates in Jk(Rn, Rm)are just P k(Rn, Rm), and when composed with projection onto the latter space,F(·, x) is just a translation (and hence a submersion) for every x.

Thus F(·, x) istransversal to M × {y}, and the hypotheses of Lemma 3 are satisfied.Theorem 3′ says, roughly speaking, that given n “codimension one” conditionson the (k −1)-jet of f (these conditions can depend on x, but none can dependonly on x) and an additional “measure zero” condition on the k-jet of f, almostevery Ck function f on a given set in Rn does not satisfy all n+1 conditions at anypoint in its domain. We now use Lemma 3 and Theorem 3′ to prove Propositions3, 7, and 8.Proof of Proposition 3.

We will show for each pair (i1, i2) with 0 ≤i1 < i2 ≤kthat almost every f in Ck(R) has the property that f (i1)(x) and f (i2)(x) are neverboth zero. Let M be the manifold in Jk−1(R, R) defined by f (i1) = 0.

Then Mhas codimension 1, and the set Z ⊂Jk(R, R) defined by f (i1) = f (i2) = 0 is a zeroset in M × bP k(R, R). Therefore by Theorem 3′, almost every f in Ck(R) has theproperty that jkf(x) is not in Z for any x ∈R, which is exactly what we wantedto prove.Proof of Proposition 7.

We first prove the proposition for fixed points using The-orem 3′. Let M be the manifold in J0(Rn, Rn) defined by f(x) = x; then x is afixed point of f if and only if j0f(x) lies in M. Notice that M has codimension n,and projection onto the first n coordinates is a submersion on M. Let Z be the setof 1-jets in M × bP 1(Rn, Rn) for which Df has an eigenvalue with absolute value1.

Then if f has a nonhyperbolic fixed point, j1f(x) must lie in Z for some x. Wewill be done once we show that Z is a zero set in M × bP 1(Rn, Rn). By Lemma2, we need only show that the set S of n × n matrices with an eigenvalue on theunit circle has measure zero (in the space of n × n matrices, which is isomorphicto bP 1(Rn, Rn)).Observe that every ray from the origin meets S in at most npoints, because multiplying every entry of a matrix by a constant factor multipliesits eigenvalues by the same factor.

Hence Z is a zero set as claimed.For orbits of period p > 1, the condition for nonhyperbolicity depends on the1-jet of f at p different points. Thus to apply here, Theorem 3′ would have to begeneralized to p-tuples of jets.

Rather than do this in general, we will prove Propo-sition 7 directly from Lemma 3. Let us determine what conditions are necessaryfor a set of polynomial functions g1, g2, .

. .

, gq : Rn →Rn to span a probe. For agiven C1 function f : Rn →Rn and λ ∈Rq, letfλ = f +qXi=1λigi.

16B. R. HUNT, TIM SAUER, AND J.

A. YORKEWe must show that for almost every λ, all periodic points of fλ with period p arehyperbolic.Let x1, x2, . .

. , xp denote elements of Rn, and let A ⊂Rnp be the set of all(x1, x2, .

. .

, xp) for which xi ̸= xj when i ̸= j. Consider the function F = (G, H),where G: Rq × A →Rnp and H: Rq × A →Rn2p are defined byG(λ; x1, x2, .

. .

, xp) = (fλ(x1) −x2, fλ(x2) −x3, . .

. , fλ(xp) −x1),H(λ; x1, x2, .

. .

, xp) = (Dfλ(x1), Dfλ(x2), . .

. , Dfλ(xp)).

(Essentially F consists of the 1-jets of f at x1, . .

. , xp, except that G projects the0-jets onto a subspace.

)For a given λ, if x1 is a point of period p for fλ, then there is a correspondingpoint (x1, . .

. , xp) ∈A at which G = 0.

We then let M = {0} in applying Lemma 3.If in addition x1 is nonhyperbolic, then the matrix Qpi=1 Dfλ(xi) has an eigenvalueon the unit circle. That is, H(λ; x1, .

. .

, xp) lies in the set S given byS =((M1, . .

. , Mp) ∈Rn2p :pYi=1Mi has an eigenvalue on the unit circle),where M1, .

. .

, Mp denote n × n matrices. As in our previous argument for fixedpoints, S has measure zero because every ray from the origin in Rn2p intersects Sin at most n points.

Thus we let Z = {0} × S in applying Lemma 3.We will be done if we can show that G and H satisfy the hypotheses of Lemma3. Now G and H satisfy the differentiability hypothesis of Lemma 3 because fλis C1 as a function of x and C∞as a function of λ.

To verify the transversalityhypothesis, we will show that for all (λ; x1, . .

. , xp) ∈Rq × A, the derivative of Fwith respect to λ has full rank.

Since F is a linear function of λ, we simply wantto show, for every (x1, . .

. , xp) ∈A, that F is onto as a function of λ.

Recall thatfλ = f + P λigi, and observe that whether or not F is onto is independent of f.We have thus reduced the problem to one of polynomial interpolation; we need onlyshow there exists a finite-dimensional vector space P of polynomial functions fromRn →Rn such that for any p distinct points x1, . .

. , xp ∈Rn and any prescribedvalues for the 1-jet of a function at the p points, there exists a function in P whose1-jet takes on the prescribed values at the prescribed points.We claim that the polynomials of degree at most 2p −1 have the above interpo-lation property.

We are referring to polynomial functions from Rn →Rn, but theinterpolation can be done separately for each coordinate in the range, so for simplic-ity we consider polynomials from Rn →R. Given distinct points x1, .

. .

, xp ∈Rn,consider the polynomialsPj(x) =pYi=1i̸=j|x −xi|2for j = 1, . .

. , p. Each Pj has degree 2p −2 and is zero at every xi except forxj, where it is nonzero.

Thus a suitable linear combination of the Pj can take onany prescribed values at x1, . .

. , xp.

Next let Pjk(x) be the kth coordinate of thefunction x 7→Pj(x)(x −xj) for k = 1, . .

. , n. Each Pjk has degree 2p −1, and bothPjk and its first partial derivatives are all zero at every xi, except that the kthpartial derivative of Pjk is nonzero at xj.

Then given a linear combination of the

PREVALENCE: A TRANSLATION-INVARIANT17Pj which takes on prescribed values at x1, . .

. , xp, adding a linear combination ofthe Pjk will not change these values, and a suitable linear combination of the Pjkcan be added to change the first partial derivatives at x1, .

. .

, xp to any prescribedvalues. This completes the proof.Proof of Proposition 8.

There are two main tasks involved in this proof.First,we must formulate conditions on the 3-jet of f which must be satisfied if f has anatypical (in the sense of violating one of the conditions in Proposition 8) Andronov-H¨opf bifurcation. Second, we must show that the set Z of 3-jets which satisfy theseconditions satisfies the hypotheses of Theorem 3′.The manifold M which will contain the 2-jet of every 3-jet in Z consists of those2-jets which satisfy the following two conditions:(a) f = 0.

(b) Dxf has zero trace and positive determinant. (Of course, these conditions really depend only on the 1-jet.

)Condition (b) isequivalent to the condition that Dxf has nonzero, pure imaginary eigenvalues.Notice that condition (a) defines a codimension 2 manifold, and adding condition(b) makes M have codimension 3.Condition (i) of Proposition 8 follows immediately from the implicit functiontheorem, since the determinant of Dxf is nonzero at the bifurcation point (µ0, x0).For condition (ii) of Proposition 8 to hold it suffices that the eigenvalues of Dxfat the fixed point (µ, x(µ)) have negative real parts for µ on one side of µ0 andpositive reals parts for µ on the other side. Since the eigenvalues of Dxf are complexconjugates in a neighborhood of (µ0, x0), each one has real part equal to half thetrace of Dxf.

Thus if condition (ii) of Proposition 8 fails, the following conditionmust hold:(c) The trace of Dxf(µ, x(µ)) has µ-derivative zero at µ0.The derivative of this trace depends on the 2-jet of f at (µ0, x0) and on x′(µ0),which in turn depends on the 1-jet of f.We wish to show that the set of 2-jets in M for which condition (c) holds is azero set in M. Since M depends only on the 1-jet of f, by Lemma 2 it suffices tofix the 1-jet and show that as the second derivatives in the 2-jet vary, condition(c) fails almost everywhere. Notice that fixing the 1-jet also fixes x′(µ0).

Let thecoordinates of f be (g, h) and the coordinates of x be (y, z). Then condition (c)can be written asgµy + y′(µ0)gyy + z′(µ0)gyz + hµz + y′(µ0)hyz + z′(µ0)hzz = 0,where the partial derivatives are evaluated at (µ0, x0).

As the second derivativesvary over all real numbers, the above equation holds only on a set of measure zero(a codimension 1 subspace, in fact).We have shown that condition (c) holds only on a zero set in M. By anotherapplication of Lemma 2, the set of 3-jets which satisfy condition (c) is a zero set inM × bP 3(R2, R2).If conditions (a) and (b) hold while (c) fails, then there is a condition (d) thatthe 3-jet of f must satisfy in order for condition (iii) of Proposition 8 to fail. If

18B. R. HUNT, TIM SAUER, AND J.

A. YORKEcoordinates (u, v) are chosen in such a way thatDxf(µ0, x0) =0−ωω0(where ω is the square root of the determinant of Dxf), and g and h are thecomponents of f in this coordinate system (this is different from the definition ofg and h above), then condition (d) can be written asω(guuu + guvv + huuv + hvvv) + guv(guu + gvv)−huv(huu + hvv) −guuhuu + gvvhvv = 0.See [15] for a detailed derivation of this condition, or [6] for a more expositorydiscussion of this problem.Notice that given condition (b), Dxf can be put into antisymmetric form bya linear change of coordinates depending only on the 1-jet of f at (µ0, x0), andfurther ω is nonzero and depends only on the 1-jet of f. Writing condition (d) interms of the original coordinates would be tedious; instead we employ Lemma 2again by fixing the 2-jet of f and letting its third derivatives vary. With the 2-jetfixed, the above change of coordinates is fixed, and induces a change of coordinateson the space bP 3(R2, R2).

In terms of the new coordinates, condition (d) determinesa codimension 1 hyperplane in bP 3(R2, R2), and in particular the set on which it issatisfied has measure zero. Therefore by Lemma 2, the set of all 3-jets which satisfycondition (d) is a zero set in M × bP 3(R2, R2).To summarize, we have shown that in order for the conditions given in Propo-sition 8 to fail for a given one-parameter family of vector fields f, there must bea point (µ0, x0) at which conditions (a), (b), and at least one of (c) and (d) hold.We have shown that the manifold M ⊂J2(R2, R2) defined by conditions (a) and(b) satisfies the hypotheses of Theorem 3′, and that the set Z ⊂M × bP 3(R2, R2)on which at least one of conditions (c) and (d) holds is a zero set in this mani-fold.

Therefore by Theorem 3′, for almost every f in Ck the conditions given inProposition 8 hold.Proof of Lemma 3. We assume without loss of generality that Z is a Borel set; thenso is F −1(Z).

Let π1: Rq × Rn →Rq be projection onto the first q coordinates. Wewish to show that π1(F −1(Z)) has measure zero.

Let L = G−1(M), and for x ∈Alet Lx = L ∩(Rq × {x}) be the “x-slice” of L. By the transversality hypothesis,G(·, x) is transversal to M, and thus L ⊂Rq × Rn and Lx ⊂Rq are manifolds withthe same codimension, n, as M ⊂Rm (see p. 28 of [7]).Since L has dimension q, away from its critical points π1|L is locally a diffeo-morphism. We will show that F −1(Z) is a zero set in L; then since zero sets mapto zero sets under diffeomorphisms, π1(F −1(Z)) consists of a zero set plus possiblysome critical values of π1|L.

By the Sard theorem [26], the critical values of π1|Lhave measure zero, and hence π1(F −1(Z)) has measure zero as desired.To show that F −1(Z) is a zero set in L, we first show for all x ∈A that F −1(Z)∩Lx is a zero set in Lx. Since F(·, x) is transversal to M × {y} for all y ∈Rs, andthe tangent space TλLx is the inverse image of TG(λ,x)M under DλG, and bothtangent spaces have the same codimension, it follows that DλF maps TλLx ontoTG(λ,x)M × Rs for all (λ, x) ∈B × A.In other words, F(·, x) is a submersion

PREVALENCE: A TRANSLATION-INVARIANT19from Lx to M × Rs for all x ∈A. Since Z is a zero set in M × Rs, its preimageF −1(Z) ∩Lx is a zero set in Lx as claimed.It remains only to show that the partition {Lx} of L satisfies the hypotheses ofLemma 2.

Let π2: Rq × Rn →Rn be projection onto the last n coordinates. Foreach (λ, x) ∈L, the kernel of π2 in T(λ,x)L is just TλLx.

Since the former space hasdimension q and the latter space has dimension q −n, it follows that π2 has rankn on T(λ,x)L. Thus π2|L is a submersion, which implies (see p. 20 of [7]) that nearevery point in L there is a local C1 coordinate system on L whose last n coordinatesare the same as those of x. The slices Lx of L are parallel hyperplanes in such acoordinate system, and therefore the hypotheses of Lemma 2 are satisfied.5.

Extensions of prevalenceIn this article we have proposed sufficient conditions for a property to be saidto be true “almost everywhere”, in a measure-theoretic sense, on complete metriclinear spaces. In other contexts more general definitions may be appropriate.

Forinstance, the concepts of shyness and prevalence can be extended from vector spacesto larger classes of topological groups [17].We have concentrated thusfar on extending the notions of “measure zero” and“almost every” to infinite-dimensional spaces. We now briefly consider some waysto characterize sets which are neither shy nor prevalent in an infinite-dimensionalvector space V .Definition 10.

Let P be the set of compactly supported probability measures onthe Borel sets of V . The lower density ρ−(S) of a Borel set S ⊂V is defined to beρ−(S) = supµ∈Pinfv∈V µ(S + v).The upper density ρ+(S) is given byρ+(S) = infµ∈P supv∈Vµ(S + v).If ρ−(S) = ρ+(S), then we call this number the relative prevalence of S.One can show that for all µ, ν ∈P,infv∈V µ(S + v) ≤infv∈V µ ∗ν(S + v) ≤supv∈Vµ ∗ν(S + v) ≤supv∈Vν(S + v),and thus 0 ≤ρ−(S) ≤ρ+(S) ≤1 for all Borel sets S. It follows that a shy set hasrelative prevalence zero and a prevalent set has relative prevalence one.

However,sets with relative prevalence zero need not be shy; all bounded sets have relativeprevalence zero, for example.In Rn, having positive lower density is a much stronger condition on a set thanhaving positive Lebesgue measure. The following weaker conditions give a closeranalogue to positive measure.Definition 11.

A measure µ is said to observe a Borel set S ⊂V if µ is finite andµ(S + v) > 0 for all v ∈V . A Borel set S ⊂V is called observable if there is ameasure which observes S, and is called substantial if it is observed by a compactlysupported measure.

More generally, a subset of V is observable (resp. substantial)if it contains an observable (resp.

substantial) Borel set.

20B. R. HUNT, TIM SAUER, AND J.

A. YORKEEvery set with positive lower density is then substantial, and every substantialset is observable. As in Fact 3, if µ observes a Borel set S then so does µ∗ν for anyfinite measure ν.

It follows that an observable set is not shy. In Rn, it follows as inFact 6 that a set is observable if and only if it contains a set of positive Lebesguemeasure.

In a separable space every open set is observable; given a countable densesequence {xn}, the measure consisting of a mass of magnitude 2−n at each xnobserves each open set.AcknowledgmentsThis paper has been greatly enhanced by the input of many people, only a fewof whom we can acknowledge individually. We thank P. M. Fitzpatrick for pointingout the existence of [23], a paper on a Sard-type theorem for Fredholm operators,in which the concept of “0-preconull” is introduced.This concept is similar inprinciple to a local version of shyness using measures which have finite-dimensionalsupport.

We thank A. Kagan for bringing to our attention the results of Sudakov onthe nonexistence of quasi-invariant measures, and J. Milnor for suggesting the twoexamples from complex dynamics in the introduction. We thank Yu.

Il′yashenkofor helping to simplify our proof of Fact 8 and helping to formulate the approachtaken in §4. We thank C. Pugh for his extensive comments, in particular suggestingthe line of reasoning used for the proof of Lemma 3.

We thank H. E. Nusse and M.Barge for their helpful comments.The first author is grateful to the ONT Postdoctoral Fellowship Program (ad-ministered by the American Society for Engineering Education) for facilitating hiswork on this paper.References1. V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, New York, 1983.2.

P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer.

Math. Soc.(N.S.) 11 (1984), 85–141.3.

H. Cremer, Zum Zentrumproblem, Math. Ann.

98 (1928), 151–163.4. N. Dunford and J. T. Schwartz, Linear operators, Part 1, Interscience, New York, 1958.5.

I. V. Girsanov and B. S. Mityagin, Quasi-invariant measures and linear topological spaces,Nauchn. Dokl.

Vys. Skol.

2 (1959), 5–10. (Russian)6.

J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcationsof vector fields, Springer-Verlag, New York, 1983.7. V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Englewood Cliffs, NJ, 1974.8.

M. W. Hirsch, Differentiable topology, Springer-Verlag, New York, 1976.9. B. R. Hunt, The prevalence of continuous nowhere differentiable functions, preprint.10.

W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, NJ, 1948.11.

J. A. Kennedy and J.

A. Yorke, Pseudocircles in dynamical systems, preprint.12. K. Kuratowski, Topology, Vol.

1, Academic Press, New York, 1966.13. M. Yu.

Lyubich, On the typical behavior of the trajectories of the exponential, Russian Math.Surveys 41 (1986), no. 2, 207–208.14.Measurable dynamics of the exponential, Siberian Math.

J. 28 (1987), 780–793.15.

J. E. Marsden and M. McCracken, The H¨opf bifurcation and its applications, Springer-Verlag,New York, 1976.16. M. Misiurewicz, On iterates of ez, Ergodic Theory Dynamical Systems 1 (1981), 103–106.17.

J. Mycielski, Unsolved problems on the prevalence of ergodicity, instability and algebraicindependence, The Ulam Quarterly (to appear).

PREVALENCE: A TRANSLATION-INVARIANT2118. S. E. Newhouse, On the abundance of wild hyperbolic sets, Inst.

Hautes ´Etudes Sci. Publ.Math.

50 (1979), 101–151.19. H. E. Nusse and L. Tedeschini-Lalli, Wild hyperbolic sets, yet no chance for the coexistenceof infinitely many KLUS-simple Newhouse attracting sets, Comm.

Math. Phys.

144 (1992),429–442.20. J. C. Oxtoby, Measure and category, Springer-Verlag, New York, 1971.21.

J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation,Ann. of Math.

(2) 40 (1939), 560–566.22. C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent.

Math. 15 (1972), 1–23.23.

F. Quinn and A. Sard, Hausdorffconullity of critical images of Fredholm maps, Amer. J.Math.

94 (1972), 1101–1110.24. M. Rees, The exponential map is not recurrent, Math.

Z. 191 (1986), 593–598.25.

C. Robinson, Bifurcations to infinitely many sinks, Comm. Math.

Phys. 90 (1983), 433–459.26.

A. Sard, The measure of the critical points of differentiable maps, Bull. Amer.

Math. Soc.

48(1942), 883–890.27. T. Sauer and J.

A. Yorke, Statistically self-similar sets, preprint.28. T. Sauer, J.

A. Yorke, and M. Casdagli, Embedology, J. Statist. Phys.

65 (1991), 579–616.29. H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966.30.

C. L. Siegel, Iteration of analytic functions, Ann. of Math.

(2) 43 (1942), 607–612.31. V. N. Sudakov, Linear sets with quasi-invariant measure, Dokl.

Akad. Nauk SSSR 127 (1959),524–525.

(Russian)32.On quasi-invariant measures in linear spaces, Vestnik Leningrad. Univ.

15 (1960), no.19, 5–8. (Russian, English summary)33.

L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitelymany coexisting sinks?

Comm. Math.

Phys. 106 (1986), 635–657.34.

H. Whitney, Differentiable manifolds, Ann. of Math.

(2) 37 (1936), 645–680.35. Y. Yamasaki, Measures on infinite dimensional spaces, World Scientific, Singapore, 1985.Information and Mathematical Sciences Branch (Code R44), Naval Surface WarfareCenter, Silver Spring, Maryland 20903-5000Current address: Institute for Physical Science and Technology, University of Maryland, Col-lege Park, Maryland 20742Department of Mathematical Sciences, George Mason University, Fairfax, Virginia22030Institute for Physical Science and Technology, University of Maryland, CollegePark, Maryland 20742

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