Similarly one can define -expansiveness if is not invertible. An interesting phenomenon observed by Schwartzman states that

**Theorem.** A homeomorphism cannot be -expansive (unless is finite).

This result was reported in Gottschalk–Hedlund’s book *Topological Dynamics* (1955), and a proof was given in King’s paper *A map with topological minimal self-joinings in the sense of del Junco* (1990). Below we copied the proof from King’s paper.

**Proof.** Suppose on the contrary that there is a homeo on that is -expansive. Let be the -expansive constant of , and .

It follows from the -expansiveness that is a finite number. Pick such that whenever .

*Claim.* If , then for any .

Proof of Claim. If not, we can prolong the -string since .

Recall that a pair is said to be -proximal, if for some . The upshot for the above claim is that any -proximal pair is -indistinguishable: for all .

Cover by open sets of radius , and pick a finite subcover, say . Let be a subset consisting of distinct points. Then for each , there are two points in share the room , say , and . Pick a subsequence such that and . Clearly , and . Hence the pair is -proximal and -indistinguishable. This contradicts the -expansiveness assumption on . QED.

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Now consider the complex setting, where . Let be a map from to . Then . So this time the Jacobian becomes .

Suppose is a holomorphic map on the unit disk . Then

, the area of is .

Using polar coordinate, we have , ,

and if , and if .

So .

Let be an irreducible polynomial with integer coefficients, be an irrational number such that . Then for some .

Proof. Let . For each rational number , we have . Eliminating the denominator, we have . Each item being integral, we see that and hence . On the other hand, . Combining them, we have , where .

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Let be a probability measure on , be the product measure on . Let be the shift map on , and be the projection. We consider the induced skew product on . The (largest) Lyapunov exponent of is defined to be the value such that for -a.e. .

To apply the ergodic theory, we first assume . Then is well defined. There are cases when :

(1) the generated group is compact;

(2) there exists a finite set of lines that is invariant for all .

Furstenberg proved that the above cover all cases with zero exponent:

for all other .

1. Let be a dynamical system, be a measurable and integrable function on . We define the Birkhoff sum , and the Birkhoff average whenever the limit exits.

Lemma. Suppose for -a.e. . Then for -a.e. .

That is, the only possible growth rate of a Birkhoff sum is linear. More generally, assume is ergodic and on a subset of positive measure. (1) One can apply the above lemma to the return map of on the induced map . (2) for a.e. , and hence a.e.. So Lemma 1 applies.

Proof. WLG we assume is ergodic. Then it suffices to show that . To derive a contradiction we assume , which forces .

Intuitively, we may integrate and get if one can change with . The following proof can be found in Bochi’s notes.

Let’s consider the following set , and .

Step 1. for any , there exists and such that . Otherwise, for any and for any , there exists such that . Pick the sequence inductively by , , and partition the sum to the segments , we see that , contradicting the assumption that .

Conclusion: the set is of full measure.

Step 2. For any , for some . Then we partition the segment into short segments according to the moments that . Then . Divide both sides by and let : . Integrate both sides over : .

Note that is -invariant and containing the set . So we acctually get , and hence for any . Therefore, , which contradicts with Step 1 and hence completes the proof.

2. Let be the space of lines through the origin, and each induces naturally an action on by sending to . Then we have the map , . For each probability measure on , we define the convolution be the push-forward of the measure . Then is said to be -stationary, if .

Lemma. Any -stationary measure is non-atomic.

To show , it sufficient to show that for some -stationary measure . Then a sufficient condition is

Lemma. for for all but one and -a.e. .

The line actually gives the stable bundle of at .

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(1) If is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. such that . Then we have , and , and .

(2) Suppose there exists a vector field on such that its Lie-derivative (notice the difference with ). Then Cartan’s formula says that , where . So is exact, and .

A hypersurface is of restricted contact type if is a contact form on . For example, let be a smooth function, be its Hamiltonian vector field, be a regular level set of . Then is of restricted contact type if on .

Given a time-periodic Hamiltonian on , its Hamiltonian v.f. is said to be Reeb-like if the one-form defines a contact form on the -dimensional manifold . In this case one can compute the kernel :

for all .

This is equivalent to , .

So . Then being a contact form on is equivalent to that

.

Let be a Riemannian metric on , be the Gauss curvature. Then Gauss-Bonnet Formula gives that . So if , then and hence is a flat metric. Hopf generalized this argument and proved that if has no conjugate point, then is flat.

The starting point is the geodesic flow on and the induced Ricatti equation: along a geodesic . For each , let be the solution with (the geodesic variation focuses backward at time ). By the assumption of non-conjugate point, is defined for all . Consider the limit as . This function describes the geodesic curvature of the unstable horocycle at $(x,v)$, and satisfies the following cocycle condition: . Note that the limit function $u$ is measurable, but may not be continuous.

Then we integrate over , and note that since the geodesic flow preserves the measure; .

Therefore, , and hence : the metric is flat.

Riemannian manifold , symplectic manifold , (almost) complex manifold , holomorphic tangent space . A Hermitian metric on a complex vector bundle over a smooth manifold is a smoothly varying positive-definite Hermitian form on each fiber . A (almost) Hermitian manifold is a (almost) complex manifold with a Hermitian metric on its holomorphic tangent space . Every (almost) complex manifold admits a Hermitian metric.

The real part of defines a Riemannian metric on , while the (minus) imaginary part of defines a (nondegenerate) 2-form on , the fundamental form. Any one of the three forms , , and uniquely determine the other two: , and .

If the fundamental class of a Hermitian manifold is closed, then it is called a Kahler manifold. In this case the form is called a Kahler form. A Kahler form is a symplectic form, and so Kahler manifolds are naturally symplectic.

An almost Hermitian manifold with closed is naturally called an almost Kahler manifold. Any symplectic manifold admits a compatible almost complex structure making it into an almost Kahler manifold. In holomorphic local coordinates , any Kahler form can be written as , where is a real function, the so-called Kahler potential.

As a compact manifold, admits no symplectic structure since any closed 2-form must be exact (). Note that has a well-known almost complex structure , which comes from the octonions, when the 6-sphere is viewed as the unit sphere in the 7-space of imaginary octonions. This, however, is not integrable (by the non-associativity of the octonions). An open question is whether admits a complex structure.

Kodaira and Thurston constructed a manifold that admits a symplectic form and a complex structure , which fails to be Kahler. In particular, and are not compatible. The construction is quite simple. Consider the direct product , where is the 3D Heisenberg group, be the integer lattice in , and .

Let be the natural basis of the Lie algebra , and be the natural basis of its dual . Then only non-zero bracket is , and the only non-zero differential is .

Let . Clearly the corresponding left-invariant 2-form on is nondegenerate and closed. Note that the more natural 2-form is not closed.

Let be the almost complex structure with and . Then the tensor vanishes. For example:

;

.

Therefore, the corresponding left-invariant almost complex structure on is integrable, and is a complex manifold. One can check that is not compatible with : (although compatible with ).

Moreover, the first homology group and then the first Betti number . The odd-degree Betti numbers of any Kahler manifold are ever. So is not Kahler, and there is no compatible symplectic–complex structure on .

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Note that is continuous, and non-decreasing. However, may not be strictly increasing. In fact, if and , then is locked at for a closed interval . More precisely, if for some , then on for some ; if ; while if both happen.

Also oberve that if , then is a singelton. So assuming is not unipotent for each , the function is a Devil’s staircase: it is constant on closed intervals , whose union is dense in .

9. Let be a vector field on , be the flow induced by on . That is, . Then we take a curve , and consider the solutions . There are two ways to take derivative:

(1) .

(2) , which induces the tangent flow of .

Combine these two derivatives together:

This gives rise to an equation

Formally, one can consider the differential equation along a solution :

, . Then is called the linear Poincare map along . Suppose . Then determines if the periodic orbit is hyperbolic or elliptic. Note that the path , contains more information than the above characterization.

8. Frink’s proof a metricization theorem. Suppose a topological space is endowed with a function satisfying (a) iff ; (b) ; (c) .

Then there is a metric on induced by :

It is easy to see that defines a metric on . Moreover, it is proved that .

More generally, the third condition can be replaced by (d) there exists a positive function such that if and , then . Define a sequence inductively by , , . Clearly . Then we discretize the function : let if . Then we check that the new function satisfies (c) and induces a metric . Clearly induces the same topology on as .

7. Given a embedding of a convex sphere , let be the Gauss map, that is, for each point , is the unit outer normal vector of at . Clearly is a diffeomorphism. The Gauss curvature can be viewed as a function on the standard through , and satisfies

. (*)

Minkowski problem. Given a smooth positive function on satisfying (*).

Is there a Riemannian metric on such that ?

This 2D version was also answered in the same paper of Nirenberg in 1953. For the high dimensional case see Pogorelov in 1969 and Cheng–Yau in 1976.

6. Weyl problem. Let be a Riemannian metric on the 2-sphere with positive Gauss curvature. Does there exist an isometric embedding of into ?

H. Lewy proved in 1938 the existence under the assumption that the metric is analytic, using his results on analytic Monge-Ampere equations. Nirenberg proved the existence in 1953 using his results on strong apriori estimates for nonlinear elliptic PDE in 2D. Aleksandrov obtained a generalized solution in 1948 as a limit of polyhedra, and Pogorelov proved the regularity of this generalized solution. In 1969 Pogorelov posed and solved Weyl’s problem for embedding into a three-dimensional Riemannian manifold.

5. Let be a -dimensional Riemannian manifold, be the sectional curvature tensor induced by the Levi–Civita connection. Clearly is determined by ? To what extend does determine ? See here.

Answered here by Misha Kapovich:

(1) if and has nowhere constant sectional curvature, then a diffeomorphism preserving sectional curvature is necessarily an isometry.

Explain: specifying the sectional curvature of a metric is generally a very overdetermined problem in higher dimensions.

(2) if , then there are counter-examples. Weinstein’s argument shows that every Riemannian surface admits different metrics of the same Gauss curvature (using flow orthogonal to the gradient of the curvature function).

4. Let be a diffeomorphism on a manifold , be an -invariant expanding foliation: there exists such that for any . Then the leaf volume grows polynomially.

Let . Then for any , pick . Or equally, . Then ,

3. Hilbert's fourth problem is to find all geometries whose axioms are closest to those of Euclidean geometry for which lines are geodesics. He also provided an example, a Finsler metric on a convex body. More precisely, let be a bounded convex body. Let . Then for each vector , draw the line through in the direction of . This line intersects at two points, say (forward and backward). Then the Finsler is given by . Clearly this definition depends on the shape of and the direction of . So it may not be Riemannian. Note that the Finsler is reversible. Moreover, the geodesic distance between two points is given by , where are the points of intersections of the line from to , .

2. Suspension. Let be a group action on a manifold . Then we consider the induced action of on the product manifold , where . Consider the quotient space , on which there is a natural action that comes from shift . Clearly the later commutes with the action and desends to an action on .

1. Let be a flow on , be the time-1 map, be the set of flow-invariant probability measures, be the set of -invariant probability measures. Clearly . On the other hand, for each , the measure is flow invariant. Therefore,

.

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be the correlation function.

Let be a diffeomorphism on , be the induced map, and The new correlation function

,

where . Therefore, the two smoothly conjugate systems and have the same mixing rate.

Assuming is close to identity, we see that is also uniformly expanding, and one may derive the mixing rate of independently. However, this new rate may be different (better or worse) from . For example, could be the linear expanding ones and archive the best possible rate among its conjugate class. Could one detect this rate from itself?

In the general case, two expanding maps on are only topologically conjugate (via full shifts). So it is possible that the decay rate varies in the topologically conjugate classes.

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In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson: Theorem 1 (Burns-Masur-Wilkinson) Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold…]]>

In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:

Theorem 1 (Burns-Masur-Wilkinson)Let $latex {N}&fg=000000$ be the quotient $latex {N=M/Gamma}&fg=000000$ of a contractible, negatively curved, possibly incomplete, Riemannian manifold $latex {M}&fg=000000$ by a subgroup $latex {Gamma}&fg=000000$ of isometries of $latex {M}&fg=000000$ acting freely and properly discontinuously. Denote by $latex {overline{N}}&fg=000000$ the metric completion of $latex {N}&fg=000000$ and $latex {partial N:=overline{N}-N}&fg=000000$ the boundary of $latex {N}&fg=000000$.Suppose that:

- (I) the universal cover $latex {M}&fg=000000$ of $latex {N}&fg=000000$ is
geodesically convex, i.e., for every $latex {p,qin M}&fg=000000$, there exists an unique geodesic segmentin$latex {M}&fg=000000$ connecting $latex {p}&fg=000000$ and $latex {q}&fg=000000$.- (II) the metric completion $latex {overline{N}}&fg=000000$ of $latex {(N,d)}&fg=000000$ is
compact.- (III) the boundary $latex {partial N}&fg=000000$ is
volumetrically cusplike, i.e., for…

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Today we will define the Weil-Petersson (WP) metric on the cotangent bundle of the moduli spaces of curves and, after that, we will see that the WP metric satisfies the first three items of the ergodicity criterion of Burns-Masur-Wilkinson (stated as Theorem 3 in the previous post). In particular, this…]]>

Today we will define the Weil-Petersson (WP) metric on the cotangent bundle of the moduli spaces of curves and, after that, we will see that the WP metric satisfies the first three items of the *ergodicity criterion* of Burns-Masur-Wilkinson (stated as Theorem 3 in the previous post).

In particular, this will “reduce” the proof of the Burns-Masur-Wilkinson theorem (of ergodicity of WP flow) to the verification of the last three items of Burns-Masur-Wilkinson ergodicity criterion for the WP metric and the proof of the Burns-Masur-Wilkinson ergodicity criterion itself.

We organize this post as follows. In next section we will quickly review some basic features of the moduli spaces of curves. Then, in the subsequent section, we will start by recalling the relationship between quadratic differentials on Riemann surfaces and the cotangent bundle of the moduli spaces of curves. After that, we will introduce the Weil-Petersson and the Teichmüller metrics…

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Boris Hasselblatt and Françoise Dal’bo are organizing the event “Young mathematicians in Dynamical Systems” at CIRM (Luminy/Marseille, France) from November 25 to 29, 2013. This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by…]]>

Boris Hasselblatt and Françoise Dal’bo are organizing the event “Young mathematicians in Dynamical Systems” at CIRM (Luminy/Marseille, France) from November 25 to 29, 2013.

This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by Keith Burns and myself around the dynamics of the Weil-Petersson (WP) geodesic flow.

In our mini-course, Keith and I plan to cover some aspects of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow and, maybe, some points of our joint work with Masur and Wilkinson on the rates of mixing of WP flow.

In order to help me prepare my talks, I thought it could be a good idea to make my notes available on this blog.

So, this post starts a series of 6 posts (vaguely corresponding the 6 lectures of the mini-course) on the dynamics of…

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Let be a prob space with a -algebra . Let be a sub -algebra.

Example. on , and be the Borel -algebra. Let . Note that .

Let be a -measurable r.v. and . The conditional expectation is the unique -measurable r.v. satisfying for all , and for all .

Note that if and only if is -measurable; and if is independent of .

Let be a stationary ergodic process with stationary initial distribution . A basic problem is to find sufficient conditions on and on functions such that satisfies the central limit theorem (CLT) , where the limit variance is given by .

Let be a conservative diffeomorphism on . There are two operators: , and via for all test function .

Property. (vol-preserving) and .

Let be an increasing sequence of -algebras. Then a sequence of r.v. is called a martingale w.r.t. , if is -measurable, .

Let be a decreasing sequence of -algebras. Then a sequence of r.v. is called a reverse martingale w.r.t. , if is -measurable, for any .

Theorem. Let be a stationary ergodic sequence of (reverse) martingale differences w.r.t. . Suppose , and . Then in distribution.

Gordin: Suppose is ergodic. Consider the Birkhoff sum for some with . The time series can be approximated by martingale differences provided the correlations decay quickly enough.

Suppose there exists with , such that . Then define , and let .

Property. Let be such that is decreasing. is a reverse martingale with respect to .

Proof. Note that . Then .

Let . It remains to show . To this end, we pick an element and write it as for some . Then . This completes the proof.

Three theorems of Gordin. Let be an invertible -preserving ergodic system, and be a strictly stationary ergodic sequence.

(*)

**Theorem 1.** Suppose there exists such that , . Then (*) implies exists, and in distribution (degenerate if ).

–Mixing condition. Let .

**Theorem 2.** Suppose for some , and . Then (*) implies the conclusion of Theorem 1.

–uniform mixing condition. Let .

**Theorem 3.** Suppose and . Then (*) implies the conclusion of Theorem 1.

Cuny–Merlevede: not only the CLT, but also the ASIP holds under the above conditions.

Note that we started with an invariant measure . The operator and can be defined for all non-conservative maps. To emphasize the difference, we use . Suppose for some . Then is an absolutely continuous invariant prob. measure:

.

Then we can rewrite , in the sense that

.

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