Sinai gave a systematic method to prove the ergodicity for hyperbolic systems with singularities. Under some mild conditions, Katok and Strelcyn proved that those two foliations cover a full measure set of the space and are still absolutely continuous. However, their leaves can be arbitrarily short, and the angle between them can be arbitrarily small. Assume the singularity sets of the iterates are regular. The Sinai Theorem states that local ergodicity holds if the stable and unstable cones are relatively small while the separation of the two cones is relatively not small. Then the short stable leaves and the short unstable leaves can be used to obtain local ergodic theorem. Assume that there is a continuous invariant cone field on . Then a sufficient condition for both small cones and non-small separation of cones is that , by Liverani and Wojtkowski.

]]>Let be a sequence of positive numbers such that is convergent. Then is also convergent.

Proof. The difference between the two series is that we replace by . Naturally, we split the indices into two cases:

1). the mild ones: if . That is, .

Then it is clear that , which is finite.

2). the not-so-mild case: . It follows that . Note that

when . In fact, .

It follows that , which is also finite. QED

The second one appeared in a paper of C. Liverani and M. Wojtkowski.

Let be a sequence of positive numbers and . Then is divergent if and only if is divergent.

One can also state the convergence version.

Proof. We only need to show one direction.

Let be two indices. Note that and for all .

Therefore, as .

Therefore, is divergent. QED.

In link there is an interesting observation:

**Theorem.** If for -a.e. , then .

**Proof.** Let , , and .

Note that the complement .

So if , then for each , for each ,

there exists such that .

Pick a sequence , , , for each . Then

, , : ;

, : ;

, : ;

, : ;

Add them together: with .

Applying the assumption of the theorem, we see that . Then for some .

Let be the -th return of a typical point to the set . Then . It follows that

.

This completes the proof.

]]>Then the conditional expectation . It follows that .

Now we consider another stochastic process: let , and . The process is given by , and

We will use the max norm . Consider two successive appearances of the matrix , say one time , and the second one at tome :

,

,

,

.

So if we break the process into pairs of segments of length , , where . Then the norms of the process follow the pattern , where and . One would guess that in probability one, and hence grows subexponentially in probability one.

]]>1. This action factors through an action of .

2. There exists a 3D invariant subspace .

3. The determinant is an invariant quadratic form on , and the signature of this form is .

Let be a quadratic form on , whose isometry group is , where .

This induces an injection , and an identification between and the connected component of .

The action on passes on to the projective space . The cone is invariant, and separates into two domains: one of them is homeomorphic to a disk, which the other is a Mobius band. This induces an action of on the disk.

]]>Let be a Holder potential, which induces a transfer operator on the space of continuous functions: .

Let be the spectral radius of . Then is also an eigenvalue of , which is called the principle eigenvalue. Moreover, there exists a positive eigenfunction such that . Replacing by , we will assume .

Consider the conjugate action on the space of functional (or sign measures). There is a positive eigenmeasure such that .

We normalize the pair such that . Then the measure is a -invariant probability measure. It is called the equilibrium state of .

Two continuous functions is called cohomologous if there exists a continuous function such that

.

Let be cohomologous. Then the two operators and are different, but .

Their eigenfunctions and eigenmeasures are different, but the associated equilibrium states are the same.

To find a natural representative in the class of functions that are cohomologous to , we set . Then we have

1). . So is the eigenfunction of .

2). .

So is the eigenmeasure of .

From this point of view, we might pick as the representative of .

]]>Let . We can construct a m.c.subset , and denote the corresponding local ring by .

Let be a prime ideal of . Then is m.c. We denote the corresponding local ring by .

Let be the set of all prime ideals of . For each ideal , let . The Zariski topology on is defined that the closed subsets are exactly .

A basis for the Zariski topology on can be constructed as follows. For each , let to be the set of prime ideals not containing . Then each is open.

The points corresponding to maximal ideals are closed points in the sense that the singleton .

In the case , we see that each maximal ideal corresponds to a point . So one can interprat this as . A non-max prime ideal (a non-closed point) corresponds an affine variety , which is a closed subset in . Then is called the generic point of the varity .

2. Let be a symplectic manifold, be a Lie group acting on via symplectic diffeomorphisms. Let be the Lie algebra of . Each induces a vector field . Note that , and .

Consider the 1-form induced by the contraction . Clearly this 1-form is closed: since preserves the form .

Then the action is called weakly Hamiltonian, if for every , the one-form is exact: for some smooth function on . Although is only determined up to a constant , the constant can be chosen such that the map becomes linear.

The action is called Hamiltonian, if the map , is a Lie algebra homomorphism with respect to Poisson structure. Then and .

A moment map for a Hamiltonian -action on is a map such that for all . In other words, for each fixed point , the map from to is a linear functional on and is denoted by . Also note that . So .

]]>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.

]]>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 .

]]>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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