Let 
The foliation 
Proposition. A almost periodic foliation is recurrent.
Proof. Let 
A foliation 
2. A support function on a convex domain is the signed distance. Let 
Let 
Taking derivative with respect to 
$latex $
1. Phase transitions in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way. For example water, as it is cooled and turns into ice. Phase transitions are characterized by an order quantity (like density) that changes as a function of a parameter of the system (such as the temperature). The special value of the parameter at which the system changes its phase is the system’s critical point.
A bifurcation occurs in a dynamical systems, when a small/smooth change of the parameter values (the bifurcation parameters) of a system causes a sudden ‘qualitative’ or topological change in its behavior. For example the ‘period-doubling bifurcation’ of Logistic map, the saddle-node bifurcation.
Phase transition in dynamical systems
– the parameters $t$’s where the pressure $P(f,t\phi)$ fail to be $C^k$ for $k=0,1,\cdots,\infty,\omega$;
– the parameters $t$’s where the system $f_t:M^2\to M^2$ shifts from integrability to nonintegrability, from regular to chaotic.
, for what 
does there exist a nontrivial 
such that 
? Franks proved that for some diffeo, it reduces to a homotopy problem. So the definition:
-diffeo, if given any homeo 
on a CW-complex 
with a continuous map 
such that 
from 
, there exists a unique base-point preserving map 
homotopic to 
such that 
.
, such that 
from 
.
is torsion-free and 
is 
, the 
-periodic orbits forms a continuous family and hence all the trajectories have the same length.
, whose interior contains no invariant circles, is a Birkhoff instability region. The dynamics in an instability region
-smooth billiard table with metric positive entropy? b) How to construct a convex 
-smooth billiard table with positive metric entropy?
corresponds to an (oriented) closed polygon with 
be a homeomorphism preserving the measure-class of 
, 
be the Jacobian of 
at 
. Then for any sequence 
with 
we have 
for 
. For example 
with 
or 
with 
. This is a direct corollary of Borel-Cantelli property:
. It is easy to see 
and hence 
. So 
for all 
be two probability measures on 
and 
be its density. Then 
. Moreover 
is well-defined with respect to 
and 
, too.
when 
, 
otherwise.
. So the relative entropy is nonnegative. 
and 
, then 
and 
.
be close enough, which leads to a Holder continuous conjugate 
with 
. Ruelle found an explicit formula of 
.
be two homeomorphisms, 
, and 
. Let 
. Then the map 
gives a shifted-vector field on 
.
diffeomprhism. Then 
induces the local Banach structure and turns 
into a Banach manifold.
be the decomposition of the correction 
with respect to the hyperbolic splitting 
. Then the derivative of 
in the direction of 
is given by the vector field 
.
, 
and let 
be a subset of 
be a 
-tuple of positive integers with 
.
of type 
is an atlas on 
-fold covering of a neighborhood
in 
. 
is the set of such structures modulo isotopy relative to 
. It has a canonical structure of manifold.
of the symmetric difference of two 
-dimensional disks when one center is shifted a little bit. Let’s start with 
. So we have two intervals ![[-r,r]](http://s0.wp.com/latex.php?latex=%5B-r%2Cr%5D&bg=ffffff&fg=000000&s=0 "[-r,r]")
and ![[x-r,x+r]](http://s0.wp.com/latex.php?latex=%5Bx-r%2Cx%2Br%5D&bg=ffffff&fg=000000&s=0 "[x-r,x+r]")
. It is easy to see the symmetric difference is of length 
.
: two disks 
and 
of radius 
and center distance 
. So the angle 
satisfies 
.
and 
, which have the same area: 
. Note that the limit
.
.
. Looks like it will start with a linear term 
.
be an ellipse with 
, and 
be the tangent vector at 
. Let 
to 
.
be the arc-length parameter and 
be the curvature at 
. Alternatively we have 
. 
, or equivalently, 
. 
.
, it is easy to see that 
. 
. Taking derivatives with respect to 
. Then 
, or
. Note that 
. Therefore 
, which completes the verification.
and 
such that every 
possesses a partially hyperbolic attractor 
near 
with Gibbs u-states 
on 
, then any weak limit is a Gibbs u-state on 
be an ergodic Gibbs u-state with negative central Lyapunov exponents. Then there exist an open set 
such that 
. The analog doesn’t hold for Gibbs u-states with positive central Lyapunov exponents, since the stable and unstable directions play different roles in dissipative systems.
with fiber 
. Then every nearby point 
with Birkhoff-regular plaque 
, the intersection 
has positive leaf volume, and some point in there must be Birkhoff-regular, say 
for some 
. Then Hopf test: for any 
, 
So all Birkhoff-regular plaques lie in the same ergodic omponent.
. Then there exists a 
possesses a unique Gibbs u-state 
. Moreover 
as 
. So we say 
is stably ergodic. Since all these measures are hyperbolic, further analysis shows that 
and 
depending continuously of ![\chi(x)\cap[-\delta,\delta]=\emptyset](http://s0.wp.com/latex.php?latex=%5Cchi%28x%29%5Ccap%5B-%5Cdelta%2C%5Cdelta%5D%3D%5Cemptyset&bg=ffffff&fg=000000&s=0 "\chi(x)\cap[-\delta,\delta]=\emptyset")
, the frequency of times 
such that the size of local Pesin manifolds at 
is larger than 
.![\chi(\mu)\cap[-\delta,\delta]=\emptyset](http://s0.wp.com/latex.php?latex=%5Cchi%28%5Cmu%29%5Ccap%5B-%5Cdelta%2C%5Cdelta%5D%3D%5Cemptyset&bg=ffffff&fg=000000&s=0 "\chi(\mu)\cap[-\delta,\delta]=\emptyset")
, theand hence the set 
of points with large Pesin manifolds has positive measure: by Kac’s formula, 
.
(see Theorem 3.2 there, where 
). It is unknown if the argument can be carried out to the set of (Birkhoff) homoclinic points 
(for general invariant but nonergodic measure 
with spiraling source 
at the corners. 
for every 
, where 
of 
. We can blow up the center, identify the corresponding boundaries of two copies and reverse the flow direction on the second copy. Then the subset 
.
is well defined in 
, and the set 
is well defined in the measure-class of 
.
is bi-essentially saturated (by a density argument). Similar argument shows that every invariant subset of 
-measures, which states, for example, the conditional measures 
of 
is not only abs. cont., but also smooth: the canonical density (see 
is Holder, bounded and bounded away from zero, since ACIP is automatically a Gibbs 
for 
, or equivalently,
for 
-a.e. 
on 
be a plaque of the Pesin unstable manifold of 
with the property that 
for all 
, and the normalizing condition 
. Let 
be the induced probability on 
. Then: 
for any 
. Hence 
. In particular 
.
and 
induce probabilities and must coincide:
. Such measures are called the leafwise u-Gibbs measures.
which is minimal, but not ergodic. See also Parry’s book Topics in Ergodic Theory.
be an irrational number and 
be the irrational rotation. Let 
be a smooth function, which induces a skew-product 
, 
. Consider the following cohomological equation:
, 
.
. But viewed as a 
-valued function, the obstruction is trivial for 
if 
is an integer. (also nontrivial if 
is an irrational..)
-valued solution for some 
.
, then 
, then such 
. But 
-solution, but no continuous solution. In particular the corresponding system is minimal but not ergodic.
, a homeomorphism on a compact metric space. Let 
be the collection of 
be a continuous function. Birkhoff ergodic theorem states that the time-average 
for almost every 
is a almost every defined, measurable function.
for every 
is a singleton (say also 
is a compact interval, say ![[\underline{\phi},\overline{\phi}]](http://s0.wp.com/latex.php?latex=%5B%5Cunderline%7B%5Cphi%7D%2C%5Coverline%7B%5Cphi%7D%5D&bg=ffffff&fg=000000&s=0 "[\underline{\phi},\overline{\phi}]")
. Let’s show that the time average will fall close to this interval uniformly on 
and 
such that ![\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\phi(f^ix_k)\notin[\underline{\phi}-\delta,\overline{\phi}+\delta]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bn%7D%5Csum_%7Bi%3D0%7D%5E%7Bn_k-1%7D%5Cphi%28f%5Eix_k%29%5Cnotin%5B%5Cunderline%7B%5Cphi%7D-%5Cdelta%2C%5Coverline%7B%5Cphi%7D%2B%5Cdelta%5D&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\phi(f^ix_k)\notin[\underline{\phi}-\delta,\overline{\phi}+\delta]")
for every 
. Passing to a subsequence if necessary, we can assume that it converges to some 
, which will force ![\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\phi(f^ix_k)\to a\in[\underline{\phi},\overline{\phi}]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bn%7D%5Csum_%7Bi%3D0%7D%5E%7Bn_k-1%7D%5Cphi%28f%5Eix_k%29%5Cto+a%5Cin%5B%5Cunderline%7B%5Cphi%7D%2C%5Coverline%7B%5Cphi%7D%5D&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\phi(f^ix_k)\to a\in[\underline{\phi},\overline{\phi}]")
and contradict the choice of 
. Q.E.D.
. Denote 
and 
. Then Semiuniform Erogdic Theorem concerns one-side estimate similar to UET. It states that if 
, then there exist 
such that 
for every 
, and every 
for every 
and hence contradicts the choice of 
. Note that it suffices to show 
for each 
. From now on let’s fix 
and decompose 
for some 
. So 
. Summing over 
and divide both sides by 
, we get
. Now passing 
we get 
and hence 
. Q.E.D.
be a 
be a closed, invariant subset without fixed points, that is, the vector field of 
is (topologically) transitive and hyperbolic,
for some open neighborhood 
is dense in 
;
