User pengfei - MathOverflow

Answer by Pengfei for Hyperbolic sets

2013-05-20T05:46:08Z

Let's suppose $E^u=0$ on $\Lambda$ and $\lambda\in(0,1)$ be the contracting constant of $Df$ on $\Lambda$. Pick an open neighborhood $U\supset \Lambda$ (small enough) such that $\|Df|_{U}\|<\lambda<1$. Then pick $N$ large with $\lambda^N<1/6$.

First step is to show that, every nonwandering point $x\in\Omega(f,\Lambda)$ is a periodic point.

Pick a neighborhood $B(x,\delta)\subset U$ small enough such that we can show inductively, $f^nB(x,\delta)\subset U$ for all $n\ge1$. By the nonwandering assumption, there exists $y\in B(x,\delta/3)\cap\Lambda$ such that $f^ny\in B(x,\delta/3)$ for some $n\ge N$. In particular $$f^nB(x,2\delta/3)\subset f^nB(y,\delta)\subset B(f^ny,\delta/6)\subset B(x,\delta/2).$$ As a contracting self-map, there exists a unique point $p\in B(x,\delta/2)$ fixed by $f^n$ and every point in $B(x,\delta/2)$ is attracted to $p$ under $f^n$. Hence every point in $B(x,\delta/2)$ is wandering (unless the point is $p$ itself). Since $x$ is assumed to be nonwandering, we must have $x=p$ be a periodic point.

Secondly, we show every point $x\in\Lambda$ is a periodic point.

Pick a general point $x\in \Lambda$. Note that every point $y\in\alpha(f,x)$ is nonwandering and hence periodic. So every point lying close enough to $y$ will stay close and approximate the orbit $y$ under forward iterates (by the assumption $E^u=0$ on $\Lambda$). Since we assume $y\in\alpha(f,x)$, we can pick $x_k=f^{-n_k}x\to y$. So $d(x,y)= d(f^{n_k}(x_k),y)\le d(x_k,y)\to 0$, which implies $x=y$ is also a periodic point.

Finally, the finiteness: this also follows from $E^u=0$ on $\Lambda$, since every periodic point $x\in\Lambda$ will admits an open attracting neighborhood. If $x_k\in\Lambda\to x$, then $x\in\Lambda$ and $x_k$ will fall into the attracting neighborhood of $x$ for $k\ge K$ , which will force $x_k=x$ for all $k\ge K$.

Answer by Pengfei for Conley Theorem (or fundamental theorem of dynamical systems)

2013-05-08T02:27:08Z

There is a classical example with $NW(f)\backslash \overline{R(f)}\neq\emptyset$, the so called Bowen's eye-like attractor (see the paper by Baladi, Bonatti and Bernard: Abnormal Escape Rates from Nonuniformly Hyperbolic Sets):

Every point on the dark curve is nonwandering, but only the two corners are recurrent (in fact these two are fixed).

Why is the theorem called Fundamental? One reason is that it is the 'correct' setting of $C^1$ stability conjecture (see the discuss here).

Theorem: Let $f$ be a diffeomorphism on a closed manifold $M$. Then the following are equivalent:
1. the map $f$ is structurally stable;
2. $\mathcal{R}(f)$ is hyperbolic;
3. $f$ is $\mathcal{R}$-stable.

This modern version is very succinctly comparing to the version using the nonwandering set, which involves with no-cycle condition and transversality condition.

Also I copied a few words from the paper mentioned in Barry's comment:

'The theorem is fundamental in the sense that it deals with the basic question of the field. It is also fundamental in that it encompasses such big ideas in such a small, concise statement.'

Compared to Fundamental Theorem of Arithmetic and Fundamental Theorem of Algebra, Norton wrote:

'the space on which the dynamics take place, can be decomposed uniquely into its basic dynamical parts: points whose dynamics can be described as exhibiting a particular type of recurrence, and points which proceed in a gradient-like fashion.'

Recurrence and transience of cocycle over a dynamical system

2013-04-17T06:18:51Z

Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and $\phi_n(x)=\phi(x)+\cdots+\phi(T^{n-1}x)$ be the induced cocycle.

A point $x\in X$ is said to be $\phi$-transient if $\phi_n(x)\to\infty$ as $n\to\infty$. Otherwise $x$ is said to be $\phi$-recurrent.

It may be more appropriate to define these notations in the skew-product system $(X\times\mathbb{R},\mu\times m)$, with $(T,\phi):(x,t)\mapsto (Tx,t+\phi(x))$. But above definitions also looks natural :)

Two propositions strengthen the $\phi$-recurrence property:

(1). For $\mu$-a.e. $\phi$-recurrent point $x$, $\displaystyle \liminf_{n\to\infty}|\phi_{n}(x)|=0$.

(2). Assume $(T,\mu)$ is ergodic and $\int\phi d\mu=0$. Then $\mu$-a.e. point $x$ is $\phi$-recurrent. In particular $\displaystyle \liminf_{n\to\infty}|\phi_{n}(x)|=0$ $\mu$-a.e. $x$.

My question is, how to prove these two properties?

Difference with Birkhoff Ergodic Theorem: $\frac{\phi_n(x)}{n}\to0$, but doesn't tell how $\phi_n(x)$ behaves.

Thank you!

Just recalled how to prove the recurrence in (2):

Let $C=\max_{x\in X}|\phi(x)|>0$. Then we divide $X$ into three subsets:

(recurrent part) $x\in X_r$ if $\phi_n(x)\in[-C,C]$ infinitely often;
(positive part) $x\in X_+$ if $\phi_n(x)\ge C$ for all $n$ large;
(negative part) $x\in X_-$ if $\phi_n(x)\le -C$ for all $n$ large.

Moreover both three are invariant. We need to show $X_{\pm}$ are of zero measure. If $\mu(X_+)>0$, then $\mu(X_+)=1$ (by ergodicity). So

$$0=\int\phi d\mu=\int\phi_n d\mu=\lim_{n\to\infty}\int \phi_n d\mu\ge\int\liminf_{n\to\infty}\phi_n d\mu\ge C>0,$$ contradicts in itself and hence $\mu(X_+)=0$. Similarly we have $\mu(X_-)=0$ and hence $\mu(X_r)=1$.

So (2) follows from (1).

The relations between conservative part and conservativity

2013-04-03T03:48:11Z

I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for measure-preserving.

On the other hand, we can define the conservative part $C_f$ for general measure-class preserving maps (see below). We can also say $f$ is conservative (II) if $C_f=M$.

My question is:

given a smooth map $f:M\to M$, when could we upgrade from conservative (II) to conservativity (I) (up to a change of Riemannian metric, or to some measure $\mu\sim m$)?
More generally, when could the restriction $f|_{C_f}$ be conservative (I) (assuming $m(C_f)>0$)?

Let $(X,\mu)$ be a standard measure space and $f:X\to X$ be an isomorphism under which $\mu$ is quasi-invariant. That is, $f^\ast\mu\ll \mu$ and $\mu\ll f^\ast\mu$. A measurable set $E$ is said to be wandering if all $f^nE$, $n\in\mathbb{Z}$ are mutually disjoint.

(We may call it topologically wandering if $E$ is an open subset. So we generalize the classical definition of wandering.)

It has been proved that there exists a maximal wandering set $W$ (up to a $\mu$-null set). Then the dissipative part $D_f$ of $(X,f,\mu)$ is $D_f=\bigsqcup_{\mathbb{Z}}f^nW$. Then $C_f=X\backslash D_f$ is called the conservative part of $(X,f,\mu)$. The induced partition $\lbrace C_f,D_f\rbrace$ is called Hopf decomposition (named by Halmos?)

For example, the dissipative part is trivial if $\mu$ is probability and preserved by $f$ simultaneously.

Observation: by introducing an artificial measure $\nu=\sum_{\mathbb{Z}}f^n(\mu|_W)$, the map can be made $\nu$-preserving on the dissipative part $D_f$.

What about the conservative part? Could we make it measure-preserving with respect to some measure?
More specifically, let $M$ be a closed manifold, $f:M\to M$ be a smooth diffeomorphism (say $C^\infty$ if necessary), and $m$ be the normalized Lebesgue measure (automatically quasi-invariant). Assume $m(C_f)>0$. When could we find some $\mu\sim m|_{C_f}$ that is preserved by $f$?

Thank you!

Convexity and semicontinuity of the relative entropy function

2013-04-01T20:14:09Z

There are several different definitions of relative entropy, and some of them are not equivalent. Following is the definition we will use in this question.

Let $M$ be a closed manifold and $\mathcal{P}$ the set of Borel probability measures on $M$. Given a reference measure $\omega\in \mathcal{P}$ (usually the normalized Lebsgue measure), the relative entropy of a measure $\mu\ll\omega$ is defined as:

$$E(\mu|\omega)=\int_M\log\phi_\mu \ d\mu,$$

where $\displaystyle \phi_\mu=\frac{d\mu}{d\omega}$ is the Radon-Nykodim derivative (since we assume $\mu\ll\omega$). The integration is well defined since $\mu(\lbrace\phi_\mu=0\rbrace)=0$. For example $E(\mu|\omega)\ge0$ since

\begin{align*} E(\mu|\omega)&=\int\log\phi_\mu d\mu=\int_{\phi_\mu>0}-\log\frac{d\omega}{d\mu} d\mu \\&\ge-\log\int_{\phi_\mu>0}\frac{d\omega}{d\mu} d\mu =-\log\omega(\lbrace\phi_\mu=0\rbrace)\ge0. \end{align*}

In particular $E(\mu|\omega)=0$ implies $\mu=\omega$.

Moreover, this function is convex: for all $\mu,\nu\ll\omega$, $$E(p\mu+q\nu|\omega)\le p\cdot E(\mu|\omega)+q\cdot E(\nu|\omega).$$

As noticed by Pablo (thank you!), the above claim is indeed a direct corollary of the convexity of $h(x)=x\log x$.

A more interesting statement I want to know if that, if $\mu_n\ll\omega$ such that $\mu_n\to\mu\not$$\ll\omega$, will we always have $E(\mu_n|\omega)\to+\infty$? Thank you!

Here $\mu_n\to\mu$ in the sense that $\mu_n(f)\to\mu(f)$ for all continuous functions $f$.

The paper given by Ashok below provides an equivalent definition: $\displaystyle E(\mu|\omega)=\sup_{\alpha}\sum_{A\in\alpha}\mu(A)\log\frac{\mu(A)}{\omega(A)}$, where the supremum is taken over all finite, Borel partitions $\alpha$ with $\omega(A)>0$.

In particular if $\mu\not$$\ll\omega$, we can take open sets $A_k$ with $\mu(A_k)\ge2\delta$ and $\omega(A_k)\to 0$. So $E(\mu|\omega)=+\infty$.

Now let's make a better choice of $A_k$'s such that $\mu(\partial A_k)=0$. Then if $\mu_n\to\mu$, $\mu_n(A_k)\to\mu(A_k)$ for all $k$. Hence we can pick $n_k$ such that $\mu_{n}(A_k)\ge\delta$ for all $n\ge n_k$. So for all $n\ge n_k$, we have $$E(\mu_{n}|\omega)\ge\mu_{n}(A_k)\log\frac{\mu_{n}(A_k)}{\omega(A_k)} \ge\delta\cdot\log\frac{\delta}{\omega(A_k)}\to\infty.$$

Volume of the set of transitive points of transitive diffeomorphisms

2012-06-09T04:40:37Z

Let $M$ be a compact manifold without boundary, $f:M\to M$ be a diffeomorphism. Then $f$ is said to be (topologically) transitive if $\bigcup_{\mathbb{Z}}f^nU$ is dense for every nonempty open set $U\subset M$.

Assume $f$ is transitive and $\lbrace U_k:k\ge1\rbrace$ is a subbasis of the topology on $M$. Then define the transitive set of $f$ to be $T_f=\bigcap_{k\ge1}(\bigcup_{\mathbb{Z}}f^nU_k)$. Clearly $T_f$ is a dense $G_\delta$ subset of $M$ (so topologically large).

Let $m$ be the Lebesgue measure on $M$. My question is:

Could $T_f$ be measure-theoretically meager, say $m(T_f)=0$, for some transitive $f$?

Thank you!

Answer by Pengfei for Invariant measures and recurrent sets.

2012-07-07T05:49:20Z

Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta_o$ is the only invariant measure of $\phi_1$, and $\phi_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.

Edit: See the following paper for some flow-version examples. In particular see Proposition 1 and 2 there.

Two open sets from accessible classes of partially hyperbolic systems

2012-04-16T10:52:56Z

Let $f:M\to M$ be a partially hyperbolic diffeomorphism. Assume $f$ is dynamically coherent. In particular $E^\ast$ integrates to $\mathcal{W}^\ast$ for $\ast\in\lbrace s,u,cs,cu,c\rbrace$. See here for more information.

The accessible class of $x$, denoted by $A_f(x)$, is the set of points which can be connected by $\gamma=\gamma_1\ast\cdots\ast\gamma_k$, where each $\gamma_i:[0,1]\to M$ is a smooth path in a single stable or unstable manifolds. Now let

$\mathcal{U}(x)=A_f(W^c(x))=\bigcup_{y\in W^c(x)} A_f(y)$, and
$\mathcal{V}(x)=W^c(A_f(x))=\bigcup_{y\in A_f(x)} W^c(y)$.

We know that both $\mathcal{U}(x)$ and $\mathcal{V}(x)$ are open sets.

Note that both $A_f(\cdot)$ and $W^c(\cdot)$ induce equivalence relations on $M$. So I am curious of the following questions:

Is $\mathcal{U}(\cdot)$ an equivalence relation? What about $\mathcal{V}(\cdot)$?
Moreover, when are they equal?

Note that if the second is true, (that is, $\mathcal{U}(x)=\mathcal{V}(x)$ for all $x$), then the first one is also true. So the second one is stronger.

Thank you!

Answer by Pengfei for The Manneville-Pomeau map is topologically conjugate to a full one-sides shift on two symbols. Can you give the corresponding homeomorphism?

2012-04-18T11:48:53Z

It should be 'almost 1 to 1' conjugate to one-side shift. First take $s=0$, then $f_0(x)=2x\mod 1$ has a Markov partition $\alpha=\lbrace I_0,I_1\rbrace$, where $I_0=[0,0.5],I_1=[0.5,1]$.

The partition $\alpha_n=\alpha\vee\cdots \vee f^{1-n}\alpha$ can be indexed by $w\in\lbrace 0,1\rbrace^n$, with $f^k(A_w)\subset I_{w_k}$ for all $k=0,\cdots,n-1$.

Then the related factor map is $\pi:\lbrace 0,1\rbrace^{\mathbb{N}}\to\mathbb{T}$, $\mathbf{w}=(w_n)\to\bigcap_{n\ge1 }A_{w_0\cdots w_n}$.

It is easy to see $\pi\circ \sigma=f_0\circ \pi$, and $\pi$ is 1 to 1 expect countable many points.

For other $s$, we solve $1=f_s(x)=x+x^{1+s}$, say $x_s$. Then $\alpha_s=\lbrace I_0^s,I_1^s\rbrace$ is a Markov partition, where $I^s_0=[0,x_s],I^s_1=[x_s,1]$.

Accordingly we assign indices to elements in $\alpha_{s,n}$ for all $n$ and then let $\pi_s:\lbrace 0,1\rbrace^{\mathbb{N}}\to\mathbb{T}$, $\mathbf{w}=(w_n)\to\bigcap_{n\ge1 }A^s_{w_0\cdots w_n}$.

In the case $s=0$, we know $\pi(\mathbf{w})=\sum_{n\ge1}\frac{w_n}{2^n}$. I do not know the explicit formula of $\pi_s$ for general $s$.

Curvatures of stable and unstable manifolds

2012-02-21T06:40:16Z

Let $(M,g)$ be a closed Riemannian manifold and $f:M\to M$ be a $C^r$ ($r\ge2$) Anosov diffeomorphism, that is, there is a continuous hyperbolic splitting $TM=E^s\oplus E^u$ with respect to the Riemannian metric $g$ over $M$.

We know there exists stable and unstable foliations $\mathcal{W}^s$ and $\mathcal{W}^u$ tangent to $E^s$ and $E^u$.
Moreover, all leaves $W^s(x)$ and $W^u(x)$ are $C^r$ submanifolds for every $x\in M$.
So we can talk about the sectional/Ricci/scalar curvatures of these submanifolds (endow with the restricted metrics $g|_{W^s(x)}\text{ and } g|{W^u(x)}$).

I want to know if there are some results and references about this topic.
This should be an interesting question, and some results in decay of correlations do assume the sectional curvatures are bounded.

Maybe there are some results for

more general classes of maps (say partially hyperbolic diffeo's),
more general submanifolds (say foliations with continuous tangent $E=T\mathcal{W}$).

Thank you!

Edit: the relation between the metric $g$ and the hyperbolic splitting $TM=E^s\oplus E^u$.

The Riemannian metric $g$ induces a norm $\|\cdot\|_x$ on each tangent space $T_xM$ by $\|v\|^2_x=g_x(v,v)$ for every $v\in T_xM$.

The map $f:M\to M$ is said to be hyperbolic if there exists a continuous splitting $TM=E^s\oplus E^u$ such that

the splitting is $Df$-invariant: $D_xf(E^s_x)=E^s_{fx}$ and $D_xf(E^u_x)=E^u_{fx}$,
$E^s$ is uniformly contracting: $\|D_xf^n(v)\|_{f^nx}\le C\lambda^n_s\|v\|_x$ for every $v\in E^s_x$,
$E^u$ is uniformly expanding: $\|D_xf^n(v)\|_{f^nx}\ge C^{-1}\lambda^n_u\|v\|_x$ for every $v\in E^u_x$,

for some uniform constant $C\ge1$ and $0<\lambda_s<1<\lambda_u$.

So the hyperbolicity of the map $f$ does depend on the choice of Riemannian metric.

But being hyperbolic is not sensitive to the choice of metric. For example if $f$ is hyperbolic with respect to $(M,g)$ and we have another Riemannian metric $h$ with $C_1\cdot g_x(v,v)\le h_x(v,v)\le C_2\cdot g_x(v,v)$, then $f$ is also hyperbolic with respect to $(M,h)$ (with a possible different $C$).

The upper bound of the curvatures of stable and unstable manifolds (if exists) may depend on the choice of metrics. But the property of having bounded curvature should be independent of the choice of metrics.

Lie algebra admitting some hyperbolic automorphism is nilpotent

2012-03-12T05:30:18Z

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $\mathbb{R}$ and $\phi:\mathfrak{g}\to\mathfrak{g}$ be a Lie algebra automorphism.

Viewing $\mathfrak{g}$ as a linear space and $\phi$ a linear automorphism, we can say $\phi$ is hyperbolic if the eigenvalues of $\phi$ are disjoint from $\lbrace z\in\mathbb{C}:|z|=1\rbrace$.

Then Proposition 3.6 in Smale's paper (here) says that:

Suppose that $\phi:\mathfrak{g}\to\mathfrak{g}$ is a Lie algebra automorphism which is hyperbolic as a linear map. Then $\mathfrak{g}$ must be nilpotent.

He also mentioned the following result in (Exercise in Bourbaki with hints: Algebras de Lie, Ex. 21b, p. 124.):

Let $\mathfrak{g}$ be a finite dimensional Lie algebra having an automorphism $\phi$, no eigenvalue of which is a root of unity, then $\mathfrak{g}$ is nilpotent.

Do you have ideas how to prove these results?

Thanks!

After Vladimir Dotsenko:

$$(\phi-\lambda\gamma)[u,v]=[\phi u,\phi v]-[\lambda u,\gamma v]=[(\phi-\lambda)u,\phi v]+[\lambda u,(\phi-\gamma) v].$$

Applying above to the pair $\hat{u}=\lambda^i\phi^j(\phi-\lambda)^{a}u$ and $\hat{v}=\gamma^k\phi^l(\phi-\gamma)^{b}v$ we have $$(\phi-\lambda\gamma)[\lambda^i\phi^j(\phi-\lambda)^{a}u,\gamma^k\phi^l(\phi-\gamma)^bv]= [(\phi-\lambda)\hat{u},\phi \hat{v}]+[\lambda \hat{u},(\phi-\gamma) \hat{v}]$$ $$=[\lambda^i\phi^j(\phi-\lambda)^{a+1}u,\gamma^k\phi^{l+1}(\phi-\gamma)^bv] +[\lambda^{i+1}\phi^j(\phi-\lambda)^au, \gamma^k\phi^l(\phi-\gamma)^{b+1}v].$$

Tracing the indices we get $$(i,j,a;k,l,b)\overset{\phi-\lambda\gamma}{\to}(i,j,a+1;k,l+1,b)\cup (i+1,j,a;k,l,b+1),$$ and in particular $(a,b)\overset{\phi-\lambda\gamma}{\to}(a+1;b)\cup (a;b+1)$. Then $$(\phi-\lambda\gamma)^{m+n}[u,v] =\sum_{a+b=m+n}[\lambda^i\phi^j(\phi-\lambda)^{a}u,\gamma^k\phi^l(\phi-\gamma)^bv]=0$$ since either $a\ge m$ or $b\ge n$.

Transitivity of a flow and its time-1 map

2012-01-02T15:13:14Z

Let $M$ be a closed manifold, $f:M\to M$ be a homeomorphism, and $\phi_t:M\to M$ be a flow. .

The map $f$ is said to be (point)-transitive if some orbit $\lbrace f^nx:n\in\mathbb{Z}\rbrace$ is dense in $M$.

The flow $\phi_t$ is said to be (flow)-transitive if some flow line $\lbrace\phi_tx:t\in\mathbb{R}\rbrace$ is dense in $M$.

My first question is:

is there any sufficient condition under which the time-1 map of a transitive flow is still transitive?

In the following we assume $f$ is (point)-transitive and consider a special class of flows: suspensions of $f$.

Let $r\le R$ be positive numbers and $c:M\to[r,R]$ be a continuous suspension function. Let
$M_c=\lbrace(x,r):0\le r\le c(x),x\in M\rbrace/\sim$ where $(x,c(x))\sim(fx,0)$,
and $f_t:M_c\to M_c$ represented by the time-translation $(x,r)\mapsto(x,r+t)$.

According to our definition, the suspension flow $f_t:M_c\to M_c$ is (flow)-transitive.

If $c\equiv1$, it is easy to see that the time-1 map $f_1$ is not transitive (as a homeomorphism on $M_1$). So my second question is:

is the time-1 map $f_1:M_c\to M_c$ (point)-transitive whenever $c$ is not constant?

Any proof or reference are good. Thank you!

As suggested by Zarathustra, it is worth to point out that the time-1 map of constant suspension flow with $c=\sqrt{2}$ is also (point)-transitive.

Entropy of first return map and suspension flows

2011-04-17T10:46:36Z

There are some well know formulas of Abramov about derived systems.

Firstly let $(X,\mu,f)$ be a probability preserving system and $A\subset X$ is measurable such that $\bigcup_{n\ge0}f^nA=X$. Let $\mu_A$ be conditional probability of $(\mu,A)$ and $f_A:A\to A$ be the first return map with respect to $A$, that is, $f_A(x)=f^{k(x)}x$ where $k(x)=\inf[k\ge1: f^kx\in A]$ (well defined up to a null set). Then Abramov proved that
$h(f_A,\mu_A)\cdot\mu(A)=h(f,\mu)$.

Secondly let $(X,\mu,f)$ be a probability preserving system and $r:X\to (c,\infty)$ be a roof function with $c>0$ and $\int rd\mu<\infty$. Then consider the suspension space $X_r=[(x,y):x\in X,0\le y\le r(x)]/\sim$ where $(x,r(x))\sim(fx,0)$, the suspension measure $\tilde{\mu}$ given by $\tilde{\mu}(A)=\int_X |A_x|d\mu(x)/\int_X rd\mu$, and the suspension flow $\tilde{f}_t:X_r\to X_r,[x,y]\mapsto[x,y+t]$. Abramov also proved that
$h(\tilde{f},\tilde{\mu})\cdot\int_X rd\mu=h(f,\mu)$.

I think there are direct/intuitive proofs of these two entropy formulas. Any explanation will be great.

Thanks~

For example the first can be derived from the discrete version of the second one:

The first return time $k:A\to\mathbb{N}$ can be viewed as a discrete roof function on $A$. Then suspension is $A_k=[(x,k):x\in A,k=0,1,\cdots, k(x)]/\sim$, which is isomorphic to $X$. And the map $A_k\to A_k,[x,k]\to[x,k+1]$ is isomorphic to $T$ on $X$ (note that $\int_A k(x)d\mu(x)=1$, or equally $\int k(x)d\mu_A(x)=1/\mu(A)$). By identifying $A$ with $A\times[0]\subset A_k$, we see that
$h(f,\mu)/\mu(A)=h(f_A,\mu_A)$.

Still I have no idea about the proof about the entropy of suspension flows~

Finally I understood Abramov's proof (indeed his proof is very clear). Pick $t\in(0,c)$ (fixed from now on) and consider the subset $A=[(x,s):0\le s < t]\subset X_r$. Then the first return map $\tilde{f}_A$ of $\tilde{f}_t$ with respect to $A$ is given by $(x,s)\mapsto(fx,s-r(x))$, where $s-r(x)\in\mathbb{T}_t$ (Note that we can view $A=X\times\mathbb{T}_t$ and $\tilde{f}_A$ as a fiber extension of $f$).

He showed that $h(\tilde{f}_A,\tilde{\mu}_A)=h(f,\mu)$ (since the extension is isometric on the fiber).
As the first return map of $(X_r,\tilde{f}_t)$, $h(\tilde{f}_A,\tilde{\mu}_A)\cdot\tilde{\mu}(A)=h(\tilde{f}_t,\tilde{\mu})$.

Plugging in $\tilde{\mu}(A)=\frac{t}{\int rd\mu}$, he got the desire formula $h(\tilde{f}_t,\tilde{\mu})=h(f,\mu)\cdot\frac{t}{\int rd\mu}$.

Answer by Pengfei for Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

2012-01-02T14:49:52Z

I think the answer is positive for the special case (2).

In this case every $(X,T)$-invariant measure $\mu$ is also $(\widetilde{X},T)$-invariant and hence admits an ergodic decompostion, say $\tau=\tau_\mu$. Since $\mu(X)=1$, $m(X)=1$ for $\tau$-a.e. $m\in E(\widetilde{X},T)$. In other words, the ergodic decomposition of $\mu$ is 'localized' on $(X,T)$. Note that $h_m(X,T)=h_m(\widetilde{X},T)$ for each $m$ with $m(X)=1$. Therefore

$h_\mu(X,T)=h_\mu(\widetilde{X},T)=\int_{E(T)}h_m(\widetilde{X},T)d\tau(m)=\int_{E(T)}h_m(X,T)d\tau(m)$.

After André Caldas: I take a snapshot (link) of Theorem 6.4 in Chapter II of Mane's book:

Glue two solenoids along their boundaries

2011-01-03T14:28:12Z

Here a solenoid is a dynamical system $(N,f)$ where $N$ is the solid torus $N=\mathbb{D}^2\times S^1$ with boundary $S^1\times S^1$, and $f:N\to N$ is a smooth embedding whose image is wrapped twice in $N$. For example Smale solenoid $f(z,w)=(\frac{1}{4}z+\frac{1}{2}w,w^2)$.

I am wondering if we can glue two solenoids together to formulate a diffeomorphism. What I have in mind is consider the diffeomorphism $f:N\to fN$ and $f^{-1}:fN\to N$. I want to glue the two disjoint copies $(N,f)$ and $(fN,f^{-1})$.

The basic picture for it is to glue $g:\mathbb{D}^2\to \mathbb{D}^2,x\mapsto x/2$ with $g^{-1}:g\mathbb{D}^2\to \mathbb{D}^2$. We can add a collary to their boundaries on which $g$ and $g^{-1}$ can be glued. The result manifold is just the two-sphere $S^2$ and the map is the North--South map.

I have no idea about the solenoid situation.

Also the topological dimension of $\cap_{n\ge1}f^nN$ is 1. I also want to know if there are higher dimensional solenoids.

Thanks!

Hopf decompostion for diffeomorphisms

2011-09-04T03:41:28Z

Let $M$ be a compact smooth manifold and $m$ be a normalized volume induced by some Riemannian metric on $M$. Let $f\in\mathrm{Diff}^1(M)$ and $R_f$ be the set of recurrent points of $f$ (A point $x$ is recurrent if $x\in\omega(f,x)\cap \alpha(f,x)$, or equally $\liminf_{n\to\pm\infty}d(f^nx,x)=0$).

The Hopf decomposition of $(M,f)$ is a partition $M=C_f\sqcup D_f$ such that

the restriction $f|_{C_f}$ is conservative. That is, if $E\subset {C_f}$ is a measurable subset with $[f^nE:n\in\mathbb{Z}]$ mutually disjoint, then $m(E)=0$;
the restriction $f|_{D_f}$ is dissipative. In fact ${D_f}$ is the disjoint union of $[f^nF:n\in\mathbb{Z}]$ for some measurable subset $F\subset M$.

See here (Jon Aaronson, An introduction to infinite ergodic theory, Page 15).

By Poincare Recurrence theorem (Page 17, same book) we have $m(C_f\backslash R_f)=0$.

My question is about the converse direction:

If $m(R_f)>0$, will we have $m(C_f)>0$?
When will the following be true: $m(R_f\backslash C_f)=0$?

Accumulation points of the Birkhoff average of $m$

2011-08-31T01:57:13Z

Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by

----------$f^km(A):=m(f^{-k}A)$ for all measurable subset $A\subset M$.

Then the Birkhoff averages $\nu_k=\frac{1}{k}\sum_{j=0}^{k-1}f^jm$ are probability measures on $M$ for all $k\ge1$. The question is:

What can we say about the measure(s) in the set $\mathcal{V}(m)$ of accumulation points of $\{\nu_k:k\ge1\}$?

We know that there exists a unique SRB measure $\mu_+$ for $f$ (and a unique SRB measure $\mu_-$ for $f^{-1}$). Do we have $\mathcal{V}(m)\subset\{\mu_+,\mu_-\}$?

Extension of integrable distribution over a subset

2011-08-21T06:21:18Z

Let $M$ be a smooth manifold and $G_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G_k(M)$ be a continuous distribution on $K$.

We say $E$ is integrable on $K$ if there exists a foliation $\mathcal{F}$ (or lamination, since it may only foliates a subset of $M$), such that $T_x\mathcal{F}(x)=E_x$.

My quesion is: if $(E,K)$ is integrable, will there exist an open neighborhood $U\supset K$ that admits an integrable extension $\widetilde{E}:U\to G_k(M)$?

For example $K$ is a hyperbolic invariant set of a diffeomorphism $f:M\to M$. It is known that there are stable and unstable foliations (manifolds) through $K$. I donot know if we can extend the foliations to an open neighborhood of $K$.

fundamental domain of universal covering

2011-05-18T13:15:42Z

Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset \widetilde{M}$ such that
1. the union of $\gamma D$ over all $\gamma\in \pi_1(M)$ covers $\widetilde{M}$,
2. the collection $\gamma D^o$ are mutually disjoint,
3. $\pi(D)=M$ and the restriction $\pi|_{D^o}:D^o\to M$ is diffeomorphic onto its image.

My question is:
Does there always exist some simply connected fundamental domain?
Is every fundamental domain simply connected?

Motivation.
I saw the following statement in several papers about dynamical systems: let $B^d(0,1)$ be the unit ball in $\mathbb{R}^d$ and $M$ be a $d$-dimensional compact connected manifold without boundary. Then $M\simeq B^d(0,1)/\sim$ where $\sim $ represents some gluing along $S^{d-1}=\partial B^d(0,1)$.
I think the statement might be related to above question.

Thanks!

Is volume--preserving an intrinsic property?

2011-02-07T02:10:52Z

Let $M$ be a compact smooth manifold without boundary. A Riemannian metric $g$ on $M$ induces a volume measure (or Lebesgue measure) $m_g$ on $M$.

A diffeomorphism $f:M\to M$ is said to be {volume--preserving} if $f_*(m_g)=m_g$, that is, for each Borel subset $A\subset M$, $f_*(m_g)(A):=m_g(f^{-1}A)=m_g(A)$. Or equivalently the Jacobian (determinant of the tangent map $Df_x:T_xM\to T_{fx}M$) satisfies $J(f,m_g)(x)=1$ for every point $x\in M$.

If we change the Riemannian metric to $g'$ and the induced measure $m_{g'}$, the volume--preserving property with respect to $g$ is slightly distorted: there exists a uniform constant $C\ge1$ such that

$C^{-1}\le J(f^n,m_{g'})(x)\le C$, for every point $x\in M$ and every time $n\in\mathbb{Z}$.---- $(*)$

My question is: is $(*)$ a characterization of volume--preserving?

That is, for a given $f\in\mathrm{Diff}(M)$, if $(*)$ holds for some arbitrarily chosen Riemannian metric $g$, does there exist a Riemannian metric $g'$ such that $J(f,m_g)=1$?

Thanks!

A simple ODE on smooth manifolds

2011-01-23T00:34:44Z

For a Riemannian manifold $(M,g)$, the geodesic flow is $\phi_t:TM\to TM, (x,v)\mapsto (\gamma(t;x,v),\dot{\gamma}(t;x,v))$, where $\gamma(\cdot;x,v)$ is the geodesic started at $x$ with direction $v$. The related vector field of this flow can be formulated as $X(x,v)=(v,0)$.

My question can be viewed as the sham geodesic flow on manifolds without Riemannian metric. Namely, let consider the smooth manifold $TM$ and a vector field

$X:TM\to T(TM),(x,v)\mapsto(v,0)$.

By ODE we know this vector field can integrate to a (local) flow $\phi_t:TM\to TM$ (regardless any Riemannian metric). What is the meaning of this flow? Could we give an explicit formula of this flow?

In the simplese case $M=\mathbb{R}^d$, we know $\phi_t(x,v)=(x+t\cdot v,v)$ (indeed this is the geodesic flow with respect to the flat metric). I do not know if there is some similar form for flows on general manifolds.

Edit: Bill pointed out that the problem is not well formulated. I realized that I misunderstood the canonical splitting into horizontal and vertical parts:

$T(TM)=H\oplus V$ where the horizontal subspace $H$ is the kernel of the connection map $K : T(TM)\to TM$ and the vertical subspace $V = \mathrm{ker}(d\pi)$ is tangent to the fibers of $\pi:TM\to M$.

So I think the problem is that the splitting does not make sense if we do not have some metric at hand. And the vector field can not be defined like that.

The shape of Bowen balls

2011-01-20T03:40:03Z

These balls appear in the definition of topological entropy given by Rufus Bowen.

Let $X$ be a compact metric space, $f:X\to X$ be a homeomorphism. For each $n\ge1$ the Bowen ball $B(x,\epsilon,n)$ is given by

$B(x,\epsilon,n)=\cap_{0\le k\le n-1}f^{-k}B(f^kx,\epsilon)=(y\in X:d_n(x,y)<\epsilon),$

where $d_n(x,y)=\max_{0\le k\le n-1} d(f^kx,f^ky)$.

These new metrics $d_n$ are equivalent to $d$. So they induce the same topology on $X$. In particular for each fixed $n\ge1$, there exists $\epsilon(n)$ small enough such that all Bowen balls $B(x,\epsilon,n)$ are really 'balls'.

I think, in general, the shapes of these balls could be quite wild: the scale $\epsilon(n)$ does depend on $n$.

But in differential dynamical systems (e.g. $f\in\mathrm{Diff}(M)$), it seems that these balls have quite ideal shape: the scale $\epsilon(n)$ can be chosen uniformly, such balls are simply connected, even sort of convex.

If the map $f$ is Anosov, it is OK since we have transversal stable ans unstable foliation, Markov partitions, etc. But for general maps?

So my question is, is this really the case, that these balls are ideal balls? Thanks!

center depth of Birkhoff center

2010-12-23T11:29:50Z

I saw a statement about the Birkhoff center. Namely let $X$ be a compact metric space and $f:X\to X$ be a homeomorphism on $X$. Then for each ordinal $\alpha$ we define

for $\alpha=0$, let $\Omega_0(f)=X$,
for a successor ordinal $\alpha=\beta'$, let $\Omega_{\alpha}(f)=\Omega(f,\Omega_\beta(f))$,
for a limiting ordinal $\alpha$, let $\Omega_\alpha(f)=\bigcap_{\beta<\alpha}\Omega_\alpha(f)$.

The proposition is

For each homeomorphism $f:X\to X$ on a compact space $X$, there exists a countable ordinal $\alpha$ such that $\Omega_{\alpha'}(f)=\Omega(f,\Omega_\alpha(f))$.

This implies $\Omega_{\beta}(f)=\Omega(f,\Omega_\alpha(f))$ for all $\beta>\alpha$. The least of such $\alpha$ is called the center depth of $(X,f)$ and the corresponding $\Omega_\alpha(f)$ is called the Birkhoff center of $(X,f)$.

I tried several times to find a proof. Have you seen this before? Thanks!

Answer by Pengfei for Is there a dynamical system such that every orbit is either periodic or dense?

2010-11-27T03:01:10Z

In the following paper the authors give an almost 1-1 extension for a minimal system $(X,\mathbb{Z})$ which is transitive and the only non-transitive point is a fixed point.

For $\mathbb{N}$ action they can have a similar one with positive topological entropy.

T.Downarowicz, X. Ye: When every point is either transitive or periodic, Colloq. Math. 93 (2002) pp. 137-150.

I do not know whether hese examples can exists on manifolds.

nowhere vanishing vector field on a manifold

2010-11-24T01:42:24Z

I am wondering if there are necessary and sufficient conditions under which an one-dimensional subbundle of $TM$ has a nowhere vanishing vector field.

More precisely let $M$ be a compact smooth manifold.

a. When dose there exist a one-dimensional (smooth or continuous) subbundle $L\subset TM$?

b. If $L\subset TM$ is a continuous/smooth line subbundle of $TM$, does there exist a nowhere vanishing continuous/smooth section $X:M\to L$? If so, the euler characteristic of $L$ should be zero.

This is related to the partially hyperbolic system $f:M\to M$ and $TM=E^s\oplus E^c\oplus E^u$. I am curious if there is a 'center flow' if $\dim E^c=1$.

To Ryan: Am I right to say the following about your answers:

If there exists an 1-dimensional subbundle $L$ of $TM$, then $\chi(M)=0$. This is independent of the case whether $M$ is orientiable or not.
If $M$ is orientable, then there always exists an orientable 1-dimensional subbundle $L$ of $TM$.

Another question is, when is an 1-dimensional subbundle $L\subset TM$ orientable? Is it sufficient to assume that $M$ is orientable?

Thank you all. I did not formulate some questions properly. What I really mean is:

I. For a given line bundle $L\subset TM$, what is the obstruction for $L$ beging orientable? (or equivalently trivial according to Georges)

For example let $L_{\mathbb{C}}$ be a complex line bundle over a complex manifold $M$, if the top Chern class $c_1(L_{\mathbb{C}})$ does not vanish, then $L_{\mathbb{C}}$ can not be trivial. Is there some similar results in the real case?

II. Is there an example such that $M$ is orientable and has a non-orientable line bundle $L\subset TM$?

nonwandering set and Birkhoff center

2010-11-23T17:34:57Z

I am wondering if there are some results about the depth of a diffeomorphism on a manifold.

More precisely, $(M,f)$ be a diffeomorphism. For each compact invariant subset $E$, let $\Omega(f, E)$ be the nonwandering subset of $f$ relative to $E$. Let $\Omega_1=\Omega(f,M)$, $\Omega_{n+1}=\Omega(f,\Omega_n)$, and $\Omega_a=\cap\Omega_b$ over $b < a$ for a limit cardinal $b$... etc

So my question is; are there some conditions under which the diffeo has a finite depth, that is, $\Omega_{n+1}=\Omega_n$ for some $n$?

There are examples of topological systems with countable depths. I do not know what can happen in the smooth category. I googled and found that the depths of circle maps or interval maps are less than 2.

Thanks!

To rpotrie: I am looking for sufficient conditions on the spaces (say, manifolds) and the maps (say, the regularity) such that $f$ has finite center depth. As rpotrie mentioned, Axiom A maps (hence all Anosov) always have center depth 1, the maps with $\Omega(f)$ hyperbolic have center depth less than 2.

For example partially hyperbolicity may not be a good candidate since the direct product $f\oplus g:M\times N\to M\times N$ has transfinite center depth if one of $f$ or $g$ has.

Answer by Pengfei for Is the composition of non-wandering maps still non-wandering?

2010-11-23T17:25:25Z

I am wondering if there is an example $(f,X)$ and $n\ge2$ with $\Omega(f)\neq\Omega(f^n)$. It is interesting to know such examples.

There is an observation that for a homeo $f:X\to X$, if $x\in\omega(x,f)$, then $x\in\omega(x,f^n)$ for each $n\ge1$. The proof is:

Let $n\ge2$ be given. Note that $\omega(x,f)=\bigcup_{0\le k< n}\omega(f^kx,f^n)$. So there exists $k$ with $x\in\omega(f^kx,f^n)$.

If $k=0$ we are done.

Otherwise let $l=n-k\in[1,n-1]$. Then $f^lx\in\omega(x,f^n)$. We show inductively $f^{jl}x\in\omega(x,f^n)$ for each $j\ge1$. Since $\omega(x,f^n)$ is strictly $f^n$-invariant and $f^{nl}x\in\omega(x,f^n)$, we get $x\in\omega(x,f^n)$, too.

$f^{(j+1)l}x=f^l(f^{jl}x)\in f^l\omega(x,f^n)=\omega(f^lx,f^n)\subset\omega(x,f^n)$, where $\in$ is from induction hypothesis and $\subset$ is from the forward invariance of $\omega$-sets.

Comment by Pengfei

2013-04-18T15:32:57Z

@Ian Oops, that is a mistake. Usually $\phi_n(x)$ is unbounded for a.e. $x$ unless $\phi$ is a coboundary.

Comment by Pengfei

2013-04-17T21:18:53Z

Proposition 4.1.18 there is about the base dynamics, and doesn't evolve the cocycle $\phi$.

Comment by Pengfei

2013-04-17T21:06:23Z

@Qiaochu Yes. This is a typo. @Ian I revised my question. I want to know how to prove these two propositions.

Comment by Pengfei

2013-04-04T15:06:04Z

@r-w You might noticed that I changed the questions a little bit. By the way, could you give another answer to elaborate some of these examples?

Comment by Pengfei

2013-04-03T13:57:11Z

I did forget to make the quasi-invariance assumption. I will edit the question. I am aware the existence of the general definition and the classification. Just I am not that familiar with the general scheme... I wonder whether the thpe III can happen in my daily life (that is, diffeomorphisms) :)

Comment by Pengfei

2013-04-03T13:52:02Z

Hi! In fact your third comment is the reason why require much stronger regularity in my question: do there exist the examples of type III in the category of smooth diffeomorphisms on closed manifolds? That is, in the class of special $\mathbb{Z}$-actions on special spaces.

Comment by Pengfei

2013-04-02T16:25:33Z

The definition given there is easier to use. And the proof of lower semi-continuity is straightforward. Thank you!

Comment by Pengfei

2013-04-02T15:54:45Z

@Asaf I supplied another definition in the question, which can be used to define the relative entropy even $\mu\not$$\ll\omega$. Surely your suggestion using Lebesgue decomposition is also an important one, just may behave a little bit wilder :)

Comment by Pengfei

2013-04-02T06:03:41Z

@ashok Could you say more about this? I don't know how to use the condition $\mu\not$$\ll\omega$.

Comment by Pengfei

2013-04-02T03:21:54Z

Yes it really is! I will revise the question and ask for the another property.

Comment by Pengfei

2012-07-21T02:45:05Z

Thank you! They asked and answered above question there.

Comment by Pengfei

2012-07-08T02:27:56Z

Yes, $f(x)$ is identically 1 outside a small neighborhood of $o$ and decreasing to 0 exponentially fast as $x\to o$. Then as $n\to \infty$, more and more mass of $\frac{1}{n}\sum\limits_{k=0}^{n-1}\delta_{\phi_kx}$ is concentrated at arbitrarily small neighborhood of $o$. I think it is not easy to give a simple characterization of conditions to ensure the transitivity of $\phi_1$. Intuitively, the homogeneity of flow induced by $X$ is broken: the orbit of $\phi_1$ is 'chaotic' and spread all over the manifold.

Comment by Pengfei

2012-04-26T01:15:05Z

@ nikutaibi I am sorry about the link. There was an extra '-' there, which lead to a different page. Now It should work. Thank you!

Comment by Pengfei

2012-04-17T05:37:57Z

Yes you are right ^_^ Thank you!

Comment by Pengfei

2012-03-25T14:10:05Z

@ rpotrie Plaque Expansiveness can be (at least formally) defined whenever the center foliation exists. And this is the definition in many references. Why would we need the dynamical coherence assumption to define PE? Also does the following work: if $E^c_f$ integrates to $W^c_f$, then $W^c_f$ is normally hyperbolic. So for $g$ close to $f$, $E^c_g$ also integrates to some $W^c_g$ (close to $W^c_f$). Thank you!