Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf ·...

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Lecture 3: Exponential Complexity Steven Hurder University of Illinois at Chicago www.math.uic.edu/hurder/talks/ Steven Hurder (UIC) Dynamics of Foliations May 5, 2010 1 / 15

Transcript of Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf ·...

Page 1: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Lecture 3: Exponential Complexity

Steven Hurder

University of Illinois at Chicagowww.math.uic.edu/∼hurder/talks/

Steven Hurder (UIC) Dynamics of Foliations May 5, 2010 1 / 15

Page 2: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Hyperbolic normal forms

Recall a simple example from advanced calculus.

Let f (x) = x/2.

Let g(x) be smooth with g(0) = 0, g ′(0) = 1/2.

Then g ∼ f near x = 0. That is, for δ > 0 sufficiently small, there is asmooth map h : (−ε, ε)→ R such that h−1 ◦ g ◦ h = f (x) for all |x | < δ.

Fact: Exponentially contracting (or simply hyperbolic) maps have a singleparameter (their derivative g ′(0) at the fixed point) for their germinalconjugacy class.

For maps which are “completely flat” at the origin, where g(0) = 0,g ′(0) = 1, gk(0) = 0 for all k > 1, no such classification exists.

Moral: Complexity is Simplicity.

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Page 3: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Foliation complexity

Let F be a foliation of a compact Riemannian manifold M.

For each w ∈ M the leaf Lw containing w inherits a Riemannian metric forwhich Lw is geodesically complete.

Fix Lw and then count the number of points Gr(Lw , d) = #{Lw ∩ T }.The rate of growth of the function d 7→ Gr(Lw , d) is a measure of thecomplexity of the leaf.

Lw has exponential growth type if there exists λ > 0 and d0 ≥ 0 such that

Gr(Lw , d) ≥ exp{λ · d} , d ≥ d0

Lw has polynomial growth type if there exists m > 0 and d0 ≥ 0 such that

Gr(Lw , d) ≤ mk , d ≥ d0

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Page 4: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Subexponential complexity

Definition: A foliation F is uniformly subexponential if:

for all ε > 0, there exists Cε, dε so that for all w ∈ M,

Gr(Lw , d) ≤ Cε · exp{ε · d} , d ≥ dε

The celebrated theorem of Connes, Feldman and Weiss [1981] implies:

Theorem: If F is uniformly subexponential, then the equivalence relationit defines on the transversal space T is amenable, hence hyperfinite.

The pseudogroup action GF on T is hyperfinite if it is measurably orbitequivalent to an action of the integers Z on the interval [0, 1].

Moral: Subexponential complexity often leads to ambiguity.

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Page 5: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Expansion growth

We measure exponential complexity for pseudogroup actions, following[Bowen 1971] and [Ghys, Langevin & Walczak 1988].

Let ε > 0 and d > 0. A subset E ⊂ T is said to be (ε, d)-separated if

• for all w ,w ′ ∈ E ∩ Ti• there exists g ∈ GF with w ,w ′ ∈ Dom(g) ⊂ Ti , and ‖g‖ ≤ d

• then dT (g(w), g(w ′)) ≥ ε.• If w ∈ Ti and w ′ ∈ Tj for i 6= j then they are (ε, d)-separated by default.

The “expansion growth function” counts the maximum of this quantity:

h(GF , ε, d) = max{#E | E ⊂ T is (ε, d)-separated}

If the pseudogroup consists of isometries, for example, then applyingelements of GF does not help to separate points, so these growth functionsremain polynomial as functions of d , for all ε.

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Page 6: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Foliation geometric entropy

The function d 7→ h(GF , ε, d) measures expansion growth at distance d -sort of an integrated total exponent.

Define:

h(GF , ε) = lim supd→∞

ln {max{#E | E is (ε, d)-separated}}d

h(GF ) = limε→0

h(GF , ε)

Theorem: [GLW 1988] The quantity h(GF ) is finite if F is a C 1-foliation.

Moreover, the property h(GF ) > 0 is independent of all choices.

Theorem: If F is defined by a flow φt then h(GF ) = 2 · htop(φ1).

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Page 7: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Doubling maps have entropy ln(2) > 0

Exercise: The Hirsch foliations always have positive geometric entropy.

Solution: The holonomy pseudogroup GF of the Hirsch example istopologically semi-conjugate to the pseudogroup generated by thedoubling map z 7→ z2 on S1.

After d-iterations, the inverse map to z 7→ z2d has derivative of norm 2d

so we have a rough estimate

h(GF , ε, d) ∼ (2π/ε) · 2d

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Page 8: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Orbit growth implies entropy

For the Hirsch example, notice as we wander out the tree-like leaf, we arealso wandering around the transversal space T .

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Page 9: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Manning’s Theorem

Let B be a compact manifold of non-positive curvature.

Let M = T 1B denote the unit tangent bundle to B.

Let φt : M → M be the geodesic flow of B.

Theorem: [Manning 1976] htop(φ) = Gr(π1(B, b0))

That is, the growth rate of the volume of balls in the universal covering ofB equals the entropy.

This is actually a theorem about foliation entropy and growth rates ofleaves.

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Page 10: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Fundamental domains

The assumption that B has non-positive curvature implies that its universalcovering B is a disk, and we can “color” it with fundamental domains:

The proof of Manning’s Theorem follows from the picture.

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Page 11: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Weak stable foliations

Assume that B has uniformly negative sectional curvatures.

Let φt : M → M be the geodesic flow. Define an equivalence relation onpoints of M:

w ∼φ w ′ ⇐⇒ dM(φt(w), φt(w ′)) ≤ C for t →∞

Then defineLw = {w ′ ∈ M | w ′ ∼φ w}

Theorem: [Pugh-Shub 1974] The sets Lw form the leaves of a C 1-foliationof M. The resulting foliation is called the weak-stable foliation for φt .

1) Each leaf Lw is a C∞-immersed submanifold of M.

2) The orbits of the geodesic flow φt(w) are contained in the leaves of F .

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Page 12: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Entropy for weak stable foliations

Theorem: Let B be a compact manifold of negative curvature, and let Fbe the weak stable foliation for the geodesic flow φt . Then

h(GF ) = 2 · htop(φ1)

The proof that h(GF ) =≥ 2 · htop(φ1) is easy - we use the holonomy alonggeodesic segments to separate points.

The other estimate requires knowing about the structure of the weakstable foliations - the leaves are obtained by applying the geodesic flow tothe strong stable foliations, which are polynomial growth, so do not addany exponential complexity.

Steven Hurder (UIC) Dynamics of Foliations May 5, 2010 12 / 15

Page 13: Lecture 3: Exponential Complexityhomepages.math.uic.edu/~hurder/talks/Barcelona_20100505np.pdf · 5/5/2010  · Theorem: [Pugh-Shub 1974] The sets L w form the leaves of a C1-foliation

Entropy and chaos

Question: When is h(GF ) > 0?

• Expanding holonomy (Hirsch examples)

• Weak stable foliations (for Anosov flows)

• Ping-pong games (Resilient leaves in codimension one)

Are there other cnanoical situations where we can expect positive entropy?

For example, if F has leaves of exponential growth, doses there alwaysexist a C 1-close perturbation of F with positive entropy?

Next time, we discuss the relation between foliation entropy and theexistence of hyperbolic invariant measures for the foliation geodesic flow.

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Problemos de la dia

Monday [3/5/2010]: Characterize the transversally hyperbolic invariantprobability measures µ∗ for the foliation geodesic flow of a given foliation.

Tuesday [4/5/2010]: Classify the foliations with subexponential orbitcomplexity and distal transverse structure.

Wednesday [5/5/2010]: Find conditions on the geometry of a foliationsuch that exponential orbit growth implies positive entropy.

Thursday [6/5/2010]:

Friday [7/5/2010]:

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References

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Transactions AMS,153:401–414, 1971.

H.B. Lawson, Jr., The Quantitative Theory of Foliations, NSF Regional Conf. Board Math.Sci., Vol. 27, 1975.

A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a singletransformation, Ergodic Theory Dynamical Systems, 1:431–450, 1981.

E. Ghys, R. Langevin, and P. Walczak, Entropie geometrique des feuilletages, Acta Math.,168:105–142, 1988.

S. Hurder, Entropy and Dynamics of C1 Foliations, preprint 2000.

S. Hurder, Dynamics and the Godbillon-Vey Classes: A History and Survey, in Foliations:Geometry and Dynamics (Warsaw, 2000), World Scientific Pub., 2002:29–60.

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