rational-ergodicity &laquo; WordPress.com Tag Feed http://en.wordpress.com/tag/rational-ergodicity/ Feed of posts on WordPress.com tagged "rational-ergodicity" Mon, 20 May 2013 08:37:55 +0000 http://en.wordpress.com/tags/ en <![CDATA[CF4: law of large numbers for certain cylinder flows]]> http://matheuscmss.wordpress.com/2012/04/08/cf4-law-of-large-numbers-for-certain-cylinder-flows/ Sun, 08 Apr 2012 20:08:06 +0000 yglima http://matheuscmss.wordpress.com/2012/04/08/cf4-law-of-large-numbers-for-certain-cylinder-flows/ The present post will focus on the paper Law of large numbers for certain cylinder flows in collaboration with Patricia Cirilo and Enrique Pujals, in which we construct a class of cylinder flows that are rationally ergodic along a subsequence of iterates (and thus have explicit law of large numbers). See CF3 for the definitions.

Firstly, we remind what is known: let {T:\mathbb T\rightarrow\mathbb Z}&fg=000000 be the Haar function, defined as

\displaystyle T(x)=\left\{ \begin{array}{rl} 1\ ,&\text{ if }x\in\left[0,\dfrac{1}{2}\right)\\ &\\ -1\ ,&\text{ if }x\in\left[\dfrac{1}{2},1\right)\,. \end{array} \right. &fg=000000

As we proved in CF2, the associated cylinder flow {F}&fg=000000 is ergodic, for any irrational {\alpha}&fg=000000. With respect to rational ergodicity, the only result is from J. Aaronson and M. Keane. They proved in The visits to zero of some deterministic random walks that if {\alpha}&fg=000000 is quadratic surd, then {F}&fg=000000 is rationally ergodic. An irrational number {\alpha}&fg=000000 is quadratic surd if it satisfies a quadratic equation with integer coefficients or, equivalently, if its continued fraction expansion is pre-periodic. Specifically, they proved that {F}&fg=000000 is rationally ergodic along the sequence of denominators {(q_n)}&fg=000000 of {\alpha}&fg=000000. When {\alpha}&fg=000000 is quadratic surd, the jumps between two consecutive {q_n}&fg=000000`s are not dramatic and then one can interpolate the rational ergodicity along {(q_n)}&fg=000000 to every positive integer.

It is not known to what extent their result can be extended to every irrational basis. Indeed, for a generic {\alpha}&fg=000000, the jumps between the {q_n}&fg=000000`s are large and so it is natural first to ask about rational ergodicity along a subsequence of iterates, for some roof function (not necessarily {T}&fg=000000). Our goal is to prove that such phenomenon happens for almost every {\alpha}&fg=000000.

Theorem 1 (Cirilo-L.-Pujals) For almost every {\alpha\in\mathbb R}&fg=000000, there exists a function {\phi:\mathbb T\rightarrow\mathbb Z}&fg=000000 belonging to {L^p(\mathbb T)}&fg=000000, for every {p\ge 1}&fg=000000, such that the associated cylinder flow is rationally ergodic along a subsequence of iterates. 

Our class of {\alpha}&fg=000000 will satisfy the following properties: there exists a sequence of (non consecutive) denominators {(q_n)}&fg=000000 of {\alpha}&fg=000000 such that

  1. {q_n\|q_n\alpha\|}&fg=000000 converges to zero as {n}&fg=000000 goes to infinity, and
  2. {2q_n}&fg=000000 divides {q_{n+1}}&fg=000000.

Appendix B of the paper is devoted to prove that the above properties define a set of full Lebesgue measure.

Throughout this post we will eventually change the basis rotation and for this reason we will denote the {n}&fg=000000th Birkhoff sum of a function {\psi:\mathbb T\rightarrow\mathbb Z}&fg=000000 with respect to the irrational rotation {x\mapsto x+\alpha}&fg=000000 by {S_n(\alpha,\psi)}&fg=000000. We remind that {\nu}&fg=000000 denotes the Lebesgue measure on {\mathbb T}&fg=000000.

1. The roof function {\phi}&fg=000000

The roof function we construct is different in nature from the others used in this context. Consider a sequence {(q_n)}&fg=000000 satisfying properties 1 and 2 above and let

\displaystyle \phi(x)=\dfrac{1}{2}\sum_{j\ge 1}\Big[T(q_j(x+\alpha))-T(q_jx)\Big].&fg=000000

One can see {\phi}&fg=000000 as the limit of worser and worser coboundaries

\displaystyle \phi_n(x)=\dfrac{1}{2}\sum_{j=1}^n\Big[T(q_j(x+\alpha))-T(q_jx)\Big].&fg=000000

On one hand, the telescoping character of {\phi_n}&fg=000000 allows to easily calculate its Birkhoff sums. On the other hand, the increasing bad feature of {\phi_n}&fg=000000 is what will guarantee that {\phi}&fg=000000 has the required properties. In some sense, our construction resembles Anosov-Katok method of fast approximations developed in New examples in smooth ergodic theory, in which the authors construct differentiable maps sufficiently close to fibered maps of the torus (and, more generally, of any manifold that admits a {\mathbb T}&fg=000000-action) with exotic dynamical properties. Indeed, the referred maps are obtained as limits of periodic maps and here we will also use this perspective to prove Theorem 1.

Observe that the bigger {n}&fg=000000 is, the closer the points {q_n(x+\alpha)}&fg=000000 and {q_nx}&fg=000000 are. In particular, the (at least) exponential growth of {(q_n)}&fg=000000 guarantees that {\phi\in L^p(\mathbb T)}&fg=000000 for every {p\ge 1}&fg=000000. Furthermore, the sequence {(\phi_n)}&fg=000000 converges pointwise to {\phi}&fg=000000.

More than this, condition 2 implies that the Birkhoff sums of {\phi}&fg=000000 and {\phi_n}&fg=000000 agree up to iterate {q_{n+1}}&fg=000000 in a large subset of {\mathbb T}&fg=000000. To this purpose, we need the

Lemma 2 Let {q}&fg=000000 be a positive integer and {\beta,\gamma\in\mathbb T}&fg=000000. Then the set

\displaystyle \{x\in\mathbb T\,;\,T(qx+\beta)\not=T(qx+\gamma)\}&fg=000000

has Lebesgue measure equal to {2\|\beta-\gamma\|}&fg=000000.

Proof: First, observe that changing {x}&fg=000000 by {x-\beta/q}&fg=000000, we can assume that {\beta=0}&fg=000000. The function {x\mapsto T(qx)}&fg=000000 is {1/q}&fg=000000-periodic, with

\displaystyle T(qx)=\left\{ \begin{array}{rl} 1\ ,&\text{ if }x\in\left[0,\dfrac{1}{2q}\right)\bigcup\left[\dfrac{2}{2q},\dfrac{3}{2q}\right)\bigcup \cdots\bigcup\left[\dfrac{2q-2}{2q},\dfrac{2q-1}{2q}\right)\\ &\\ -1\ ,&\text{ if }x\in\left[\dfrac{1}{2q},\dfrac{2}{2q}\right)\bigcup\left[\dfrac{3}{2q},\dfrac{4}{2q} \right)\bigcup\cdots\bigcup\left[\dfrac{2q-1}{2q},1\right). \end{array} \right. &fg=000000

For each interval {\left[\frac{i}{2q},\frac{i+1}{2q}\right)}&fg=000000, {T(qx)}&fg=000000 is different from {T(qx+\gamma)}&fg=000000 if and only one of the discontinuities {0,1/2}&fg=000000 belong to the interval in {\mathbb T}&fg=000000 defined by the points {qx}&fg=000000 and {qx+\gamma}&fg=000000. This happens for an interval of length {\|\gamma\|/q}&fg=000000 and so, multiplying by the number {2q}&fg=000000 of these intervals, the desired assertion is proved. \Box&fg=000000

Let {\mathcal D=\{0,1/2\}}&fg=000000 be the set of discontinuities of {T}&fg=000000 and define

\displaystyle \Lambda_n=\left\{x\in\mathbb T\,;\,d(q_jx,\mathcal D)>q_j\|q_j\alpha\|\text{ for }j>n\right\}. \ \ \ \ \ (1)&fg=000000

Note that

\displaystyle d(q_j(x+k\alpha),q_jx)=\|kq_j\alpha\|=k\|q_j\alpha\|\le q_j\|q_j\alpha\|&fg=000000

whenever {j>n}&fg=000000 and {k=1,\ldots,q_{n+1}}&fg=000000. This implies that

\displaystyle S_k(\alpha,\phi)(x)=S_k(\alpha,\phi_n)(x)\ \ ,\ x\in\Lambda_n,\ k=1,\ldots,q_{n+1}.&fg=000000

The {\Lambda_n}&fg=000000`s form an ascending chain of subsets of {\mathbb T}&fg=000000 and {\mathbb T\backslash\Lambda_n}&fg=000000 has, by Lemma 2, Lebesgue measure at most {\sum_{j>n}q_j\|q_j\alpha\|}&fg=000000, which by property 1 can be assumed to be small.

2. Ergodicity

In this section we prove that, with the above definition, the cylinder flow {F}&fg=000000 is ergodic. We proceed in two steps.

Step 1. For any {A\subset\mathbb T\times\{0\}}&fg=000000 of positive measure, the union {\bigcup_{n\ge 1}F^nA}&fg=000000 contains {\mathbb T\times\{0\}}&fg=000000.

Step 2. {F(\mathbb T\times\{0\})\cap(\mathbb T\times\{1\})}&fg=000000 and {F(\mathbb T\times\{0\})\cap(\mathbb T\times\{-1\})}&fg=000000 have positive measure.

Step 2 is easy: by Lemma 2, for {s\in\{-1,1\}}&fg=000000 the set of points {x\in\mathbb T}&fg=000000 such that

  • {T(q_1(x+\alpha))=T(q_1x)+2s}&fg=000000, and
  • {T(q_j(x+\alpha))=T(q_jx)}&fg=000000 for {j>1}&fg=000000

has Lebesgue measure at least {\|q_1\alpha\|-2\sum_{j>1}\|q_j\alpha\|}&fg=000000, which can be assumed to be positive if we pass to a subsequence of {(q_n)}&fg=000000.

We now give the idea to prove Step 1. The formal proof requires care with some technicalities which I think are not worth in a first reading. The interested reader might check the details in the paper. The main observation is the following: assuming the divisibility condition on the {q_n}&fg=000000`s, the sequence {(m_n)}&fg=000000 of partial sums given by

\displaystyle \begin{array}{rcrcl} m_n&:&\mathbb T&\longrightarrow &\mathbb Z\\ &&&&\\ & & x &\longmapsto &T(q_1x)+\cdots+T(q_nx) \end{array} &fg=000000

defines a simple random walk in {{\mathbb Z}}&fg=000000. This is easy to see: each map {x\mapsto T(q_jx)}&fg=000000 is equal to 1 in {q_j}&fg=000000 intervals of length {1/2q_j}&fg=000000 each, and {-1}&fg=000000 in other {q_j}&fg=000000 intervals of length {1/2q_j}&fg=000000 each. We call each of these {2q_j}&fg=000000 intervals a plateau of order {j}&fg=000000 and denote by {I_j(x)}&fg=000000 the one that contains {x}&fg=000000. By the divisibility property, each {I_j(x)}&fg=000000 divides itself into {q_{j+1}/2q_j}&fg=000000 plateaux of order {j+1}&fg=000000, half of them on which {T(q_{j+1}x)=1}&fg=000000 and half on which {T(q_{j+1}x)=-1}&fg=000000. This proves the random walk character of {(m_n)}&fg=000000. In particular, it has the level-crossing property: almost every two random walks intersect infinitely often. Before going into the proof of ergodicity, we make a further remark: by the very definition of a plateau,

\displaystyle y\in I_n(x)\ \Longrightarrow\ m_n(x)=m_n(y).&fg=000000

Now let {B\subset\mathbb T\times\{0\}}&fg=000000 be a subset of positive measure, and let {x}&fg=000000 and {y}&fg=000000 be points of density of {A}&fg=000000 and {B}&fg=000000, respectively. The goal is to find {k}&fg=000000 such that the {k}&fg=000000th iterate of a neighborhood {I}&fg=000000 of {x}&fg=000000 is mapped into a neighborhood of {y}&fg=000000. This is equivalent to having, for every {x'\in I}&fg=000000,

  • {x'+k\alpha}&fg=000000 is close to {y}&fg=000000, and
  • {S_k(\alpha,\phi)(x')=0}&fg=000000.

As the {\Lambda_n}&fg=000000`s converges to {\mathbb T}&fg=000000, we might assume that {x}&fg=000000 is a point of density of {B\cap\Lambda_n}&fg=000000 for large {n}&fg=000000 and, whenever {k\le q_{n+1}}&fg=000000, the second condition is equivalent to the simpler one

  • {S_k(\alpha,\phi_n)(x')=0}&fg=000000.

This, in turn, means that {m_n(x'+k\alpha)=m_n(x')}&fg=000000. But, because of the first condition, very likely we will have {x'+k\alpha\in I_n(y)}&fg=000000 and thus {m_n(x'+k\alpha)=m_n(y)}&fg=000000. This hints us to do the following: let {n}&fg=000000 large such that

\displaystyle m_n(x)=m_n(y).&fg=000000

Then let {k\le q_{n+1}}&fg=000000 such that {x+k\alpha}&fg=000000 is close enough to {y}&fg=000000 to guarantee that {I}&fg=000000 is mapped inside {I_n(y)}&fg=000000. In this way, for any {x'\in I}&fg=000000,

\displaystyle m_n(x'+k\alpha)=m_n(y)=m_n(x)=m_n(x')&fg=000000

and so {I\times\{0\}}&fg=000000 is mapped, under {F}&fg=000000, to a neighborhood of {(y,0)}&fg=000000. This concludes the proof of Step 1 and thus of ergodicity.

Observe that, because we can always restrict {(q_n)}&fg=000000 to a subsequence, it is no loss of generality to assume that each orbit {x,x+\alpha,\ldots,x+q_{n+1}\alpha}&fg=000000 is {\varepsilon_n}&fg=000000-dense in {\mathbb T}&fg=000000, where {\varepsilon_n>0}&fg=000000 depends only on {q_n}&fg=000000, and so the term “close enough” in the previous paragraph makes sense.

3. Counting the number of returns to zero

The goal of this section is to calculate the number of returns of an arbitrary point {(x,0)\in\mathbb T\times\{0\}}&fg=000000 to {\mathbb T\times\{0\}}&fg=000000, under successive iterations of {F}&fg=000000. More specifically, we want to calculate the distribution of the map {R_{n+1}:\mathbb T\rightarrow\mathbb N}&fg=000000 given by

\displaystyle R_{n+1}(x)=\#\{0\le k<q_{n+1}\,;\,S_k(\alpha,\phi)(x)=0\}.&fg=000000

Once we have such information, we will be able to prove in the next section that {F}&fg=000000 is rationally ergodic along the sequence of iterates {(q_n)}&fg=000000, by showing that {\mathbb T\times\{0\}}&fg=000000 is a sweep-out set for which the Renyi inequality holds (see Definition 6 in CF3).

Firstly, to the purpose of iterates up to order {q_{n+1}}&fg=000000, we can work with {\phi_n}&fg=000000 instead of {\phi}&fg=000000. Better than this, we consider the “rational” truncated versions of {\phi}&fg=000000 defined by

\displaystyle \tilde\phi_n(x)=\dfrac{1}{2}\sum_{j=1}^n\Big[T_j(x+\alpha_{n+1})-T_j(x)\Big],&fg=000000

where {\alpha_{n+1}=p_{n+1}/q_{n+1}}&fg=000000 is the good rational approximation associated to {q_{n+1}}&fg=000000. This reduction is, again, similar in spirit to the Anosov-Katok method of fast approximations, in which the authors define transformations as the limit of coboundaries, not from the proper irrational rotation, but from good rational approximations of it. We claim that {S_k(\alpha,\phi_n)}&fg=000000 and {S_k(\alpha_{n+1},\tilde\phi_n)}&fg=000000 coincide in a large set for {k\le q_{n+1}}&fg=000000. Just define, similarly to (1), the set {\tilde\Lambda_n}&fg=000000 by the relations

  • {T(q_j(x+k\alpha))=T(q_j(x+k\alpha_{n+1}))}&fg=000000 for {j=1,\ldots,n}&fg=000000 and {k=1,\ldots,q_{n+1}}&fg=000000, and
  • {d(q_jx,\mathcal D)>q_j\|q_j\alpha\|}&fg=000000 for {j>n}&fg=000000.

By Lemma 2, the complement of {\tilde\Lambda_n}&fg=000000 has {\nu}&fg=000000-measure at most

\displaystyle \sum_{1\le k\le q_{n+1}\atop{1\le j\le n}}\|kq_j(\alpha-\alpha_{n+1})\|+\sum_{j>n}q_j\|q_j\alpha\|< |\alpha-\alpha_{n+1}| q_{n+1}^2\sum_{j=1}^n q_j+\sum_{j>n}q_j\|q_j\alpha\|,&fg=000000

which goes to zero as {n}&fg=000000 goes to infinity (by again, if necessary, passing to a subsequence of {(q_n)}&fg=000000). So the proof boils down to understanding the map {\tilde R_{n+1}:\mathbb T\rightarrow\mathbb N}&fg=000000 given by

\displaystyle \begin{array}{rcl} \tilde R_{n+1}(x)&=&\#\{0\le k<q_{n+1}\,;\,S_k(\alpha_{n+1},\tilde\phi_n)(x)=0\}\\ & &\\ &=&\#\{0\le k<q_{n+1}\,;\,m_n(x+k\alpha_{n+1})=m_n(x)\}. \end{array} &fg=000000

For each {k}&fg=000000, the value of {m_n(x+k\alpha_{n+1})}&fg=000000 is defined by the vector

\displaystyle v_k(x)=(T(q_1(x+k\alpha_{n+1})),\ldots,T(q_n(x+k\alpha_{n+1})))\in\{-1,1\}^n.&fg=000000

We claim that the possible combinatorics of this vector, as {k}&fg=000000 runs over the set {\{0,1,\ldots,q_{n+1}-1\}}&fg=000000, are equally distributed, for almost every {x\in\mathbb T}&fg=000000.

Proposition 3 For almost every {x\in\mathbb T}&fg=000000, the following holds: for each {v\in\{-1,1\}^n}&fg=000000,

\displaystyle \#\{0\le k<q_{n+1}\,;\,v_k(x)=v\}=\dfrac{q_{n+1}}{2^n}\,\cdot&fg=000000

The reader should interpret the above result as a binary tree: if we let, for {s_1,\ldots,s_j\in\{-1,1\}}&fg=000000, the set

\displaystyle B_{(s_1,\ldots,s_j)}=\{0\le k<q_{n+1}\,;\,T(q_i(x+k\alpha_{n+1}))=s_i,\ i=1,\ldots,j\},&fg=000000

then

\displaystyle B_{(s_1,\ldots,s_j)}=B_{(s_1,\ldots,s_j,1)}\sqcup B_{(s_1,\ldots,s_j,-1)},&fg=000000

and each of the sets {B_{(s_1,\ldots,s_j,1)},B_{(s_1,\ldots,s_j,-1)}}&fg=000000 has half of the cardinality of {B_{(s_1,\ldots,s_j)}}&fg=000000.

Once the lemma is established, we know exactly what is {\tilde R_{n+1}}&fg=000000.

Corollary 4 For each {m\in\{-n,\ldots,n\}}&fg=000000 with the same parity of {n}&fg=000000,

\displaystyle \tilde R_{n+1}(x)=\dfrac{q_{n+1}}{2^n}\cdot{n\choose\frac{n+m}{2}}&fg=000000

for a set of measure equal to {\frac{1}{2^n}{n\choose\frac{n+m}{2}}}&fg=000000.

We leave the proof of the corollary as an exercise: just use the random walk character of {(m_n)}&fg=000000. We now proceed to prove Proposition 3: the idea is to interpret, for each {j=1,\ldots,n}&fg=000000, the map {k\mapsto T(q_j(x+k\alpha_{n+1}))}&fg=000000 as a function in {\mathbb R}&fg=000000. Define

\displaystyle \begin{array}{rcrcl} \psi_j&:&{\mathbb R}&\longrightarrow &\{-1,1\}\\ &&&&\\ & &z &\longmapsto &T(q_jx + zq_j\alpha_{n+1})\\ \end{array} &fg=000000

Each {\psi_j}&fg=000000 is a periodic function with period

\displaystyle \dfrac{u_j}{v}=\dfrac{q_{n+1}/q_j}{p_{n+1}}\,\cdot&fg=000000

Dilating {\psi_j}&fg=000000 to {\tilde\psi_j:\mathbb R\rightarrow\{-1,1\}}&fg=000000 given by

\displaystyle \tilde\psi_j(z)=\psi_j(z/v),&fg=000000

then each {\tilde\psi_j}&fg=000000 is periodic with period {u_j}&fg=000000 and to analyse {\psi_1,\ldots,\psi_n}&fg=000000 along {\{0,1,\ldots,q_{n+1}-1\}}&fg=000000 is the same as analysing {\tilde\psi_1,\ldots,\tilde\psi_n}&fg=000000 along {\mathcal R=\{0,v,\ldots,(q_{n+1}-1)v\}}&fg=000000. Observing that {\mathcal R}&fg=000000 is a complete residue system modulo {q_{n+1}}&fg=000000, Proposition 3 is thus a consequence of the following

Lemma 5 Let {\tilde\psi_j:{\mathbb R}\rightarrow{\mathbb R}}&fg=000000 be a periodic function with period {u_j\in{\mathbb Z}}&fg=000000, {j=1,\ldots,n}&fg=000000. Assume that

  • {u_n}&fg=000000 is even and {u_j}&fg=000000 is a multiple of {2u_{j+1}}&fg=000000 for {j=1,\ldots,n-1}&fg=000000, and
  • there are {z_1,\ldots,z_n\in{\mathbb R}\backslash{\mathbb Q}}&fg=000000 such that

    \displaystyle \tilde\psi_j|_{\left[z_j,z_j+\frac{u_j}{2}\right)}\equiv 1\ \text{ and }\ \tilde\psi_j|_{\left[z_j+\frac{u_j}{2},z_j+u_j\right)}\equiv -1&fg=000000

    for {j=1,\ldots,n}&fg=000000.

Let {\mathcal R}&fg=000000 be a complete residue system modulo {r_1}&fg=000000. Then, for any sequence {(s_1,\ldots,s_n)\in\{-1,1\}^n}&fg=000000,

\displaystyle \#\{k\in\mathcal R\,;\,\tilde\psi_j(k)=s_j\text{ for }j=1,\ldots,n\}=\dfrac{u_1}{2^n}\,\cdot&fg=000000

The proof of the above lemma is by induction on {n}&fg=000000. Here we prove the case {n=1}&fg=000000 and refer the reader to the paper for the full proof. Firstly, let us give an idea of why this must be true. If, instead of being interested in the behavior of {\tilde\psi_1,\ldots,\tilde\psi_n}&fg=000000 along integers, we want to calcule the Lebesgue measure of a set with a specific combinatorics and {x=0}&fg=000000, then the result is clear. In our case, once we understand how the residue classes of {\mathcal R}&fg=000000 modulo {u_1,\ldots,u_n}&fg=000000 are related, the induction argument holds without further problems, whenever the values {q_jx+k/u_j}&fg=000000, {k\in\mathcal R}&fg=000000, are not discontinuities of {\tilde\psi_j}&fg=000000 (this is guaranteed by the assumption that {z_j\not\in\mathbb Q}&fg=000000, and it holds for almost every {x\in\mathbb T}&fg=000000).

We now prove the case {n=1}&fg=000000: let {\psi:{\mathbb R}\rightarrow{\mathbb R}}&fg=000000 be a function with period {u\in{\mathbb Z}}&fg=000000 such that

  • {u}&fg=000000 is even and
  • there is {z\in{\mathbb R}\backslash{\mathbb Q}}&fg=000000 such that

    \displaystyle \psi|_{\left[z,z+\frac{u}{2}\right)}\equiv 1\ \text{ and }\ \psi|_{\left[z+\frac{u}{2},z+u\right)}\equiv -1,&fg=000000

and let {\mathcal R}&fg=000000 be a complete residue system modulo {u}&fg=000000. We claim

\displaystyle \#\{k\in\mathcal R\,;\,\psi(k)=1\}=\#\{k\in\mathcal R\,;\,\psi(k)=-1\}=\dfrac{u}{2}\,\cdot \ \ \ \ \ (2)&fg=000000

To this matter, consider the sets

\displaystyle \begin{array}{rcl} \Psi_{+}&=&\left\{i\in\mathbb Z\,;\,i\in\left[z,z+\dfrac{u}{2}\right)\right\}\hspace{.6cm}\pmod u \ \ \text{and}\\ &&\\ \Psi_{-}&=&\left\{i\in\mathbb Z\,;\,i\in\left[z+\dfrac{u}{2},z+u\right)\right\}\pmod u\,. \end{array} &fg=000000

It is clear that {\Psi_{+}\cup\Psi_{-}=\mathbb Z/u\mathbb Z}&fg=000000 and that {\#\Psi_{+}=\#\Psi_{-}=u/2}&fg=000000. Also, {\psi(k)=1}&fg=000000 if and only if {k\equiv i\pmod u}&fg=000000 for some {i\in\Psi_{+}}&fg=000000. Because {\mathcal R}&fg=000000 is a complete residue system module {u}&fg=000000, (2) is proved.

4. Renyi inequality

It is a simple task to check that {(\tilde R_{n+1})}&fg=000000 satisfies a Renyi inequality. Here, we denote the asymptotic relation {f(n)=O(g(n))}&fg=000000 by {f\lesssim g}&fg=000000. On one hand,

\displaystyle \begin{array}{rcl} \left\|\tilde R_{n+1}\right\|_1&=&\displaystyle\int_{\mathbb T}\tilde R_{n+1}d\nu\\ &&\\ &=&\displaystyle\sum_{-n\le m\le n\atop{m\equiv n(\text{mod }2)}}\left[\dfrac{q_{n+1}}{2^n}{n\choose\frac{n+m}{2}}\right]\cdot \left[\dfrac{1}{2^n}{n\choose\frac{n+m}{2}}\right]\\ &&\\ &=&\dfrac{q_{n+1}}{2^{2n}}\displaystyle\sum_{i=0}^n{n\choose i}^2\\ &&\\ &=&\dfrac{q_{n+1}}{2^{2n}}\displaystyle {2n\choose n}\\ &&\\ &\sim&\displaystyle \frac{q_{n+1}}{2^{2n}}\cdot\displaystyle \frac{2^{2n}}{\sqrt{\pi n}}\\ &&\\ &=&\dfrac{q_{n+1}}{\sqrt{\pi n}}\ , \end{array} &fg=000000

where in the fifth passage we used Stirling`s approximation formula. On the other hand,

\displaystyle \begin{array}{rcl} \left\|\tilde R_{n+1}\right\|_2^2 &=&\displaystyle\sum_{-n\le m\le n\atop{m\equiv n(\text{mod }2)}}\left[\dfrac{q_{n+1}}{2^n}{n\choose\frac{n+m}{2}}\right]^2 \cdot\left[\dfrac{1}{2^n}{n\choose\frac{n+m}{2}}\right]\\ &&\\ &=&\dfrac{q_{n+1}^2}{2^{3n}}\displaystyle \sum_{i=0}^n {n\choose i}^3\\ &&\\ &\le&\dfrac{q_{n+1}^2}{2^{3n}}\displaystyle {n\choose\frac{n}{2}}\displaystyle\sum_{i=0}^n{n\choose i}^2\\ &&\\ &=&\dfrac{q_{n+1}^2}{2^{3n}}\displaystyle {n\choose\frac{n}{2}}\displaystyle {2n\choose n}\\ &&\\ &\sim&\sqrt{2}\cdot\dfrac{q_{n+1}^2}{\pi n} \end{array} &fg=000000

and therefore

\displaystyle \dfrac{\left\|\tilde R_{n+1}\right\|_2}{\left\|\tilde R_{n+1}\right\|_1}\lesssim \dfrac{\sqrt[4]{2}\cdot\dfrac{q_{n+1}}{\sqrt{\pi n}}}{\dfrac{q_{n+1}}{\sqrt{\pi n}}}\lesssim 1\,.&fg=000000

Finally, we prove the Renyi inequality for {(R_{n+1})}&fg=000000. Note that

\displaystyle \left|\|R_{n+1}\|_1-\|\tilde R_{n+1}\|_1\right| \le\int_{\mathbb T\backslash\tilde\Lambda_n}|R_{n+1}-\tilde R_{n+1}|d\nu \le q_{n+1}\nu(\mathbb T\backslash\tilde\Lambda_n)\lesssim \|\tilde R_{n+1}\|_1&fg=000000

and thus {\|R_{n+1}\|_1\lesssim\|\tilde R_{n+1}\|_1}&fg=000000. Similarly, {\|\tilde R_{n+1}\|_2\lesssim\|R_{n+1}\|_2}&fg=000000 and so

\displaystyle \|R_{n+1}\|_1\lesssim\|\tilde R_{n+1}\|_1\lesssim\|\tilde R_{n+1}\|_2\lesssim\|R_{n+1}\|_2\,,&fg=000000

which concludes the proof.

Previous posts: CF0, CF1, CF2, CF3.

]]> <![CDATA[CF3: infinite ergodic theory]]> http://matheuscmss.wordpress.com/2012/03/29/cf3-infinite-ergodic-theory-2/ Thu, 29 Mar 2012 15:20:38 +0000 yglima http://matheuscmss.wordpress.com/2012/03/29/cf3-infinite-ergodic-theory-2/ This post intends to treat the classical results on infinite ergodic theory, specifically Hopf’s ratio ergodic theorem, the non-existence of Birkhoff’s theorem, Aaronson’s theorem, rational ergodicity and law of large numbers. We won’t provide proofs for these results and for that we refer the reader to the book An Introduction to Infinite Ergodic Theory of Jon Aaronson.

Let {(X,\mathcal A,\mu,F)}&fg=000000 be a measure-preserving system: {(X,\mathcal A,\mu)}&fg=000000 is a measure space, {\mu}&fg=000000 a {\sigma}&fg=000000-finite measure and {F}&fg=000000 is a measurable transformation on {X}&fg=000000 that is invariant under {\mu}&fg=000000. Whenever {\mu(X)<\infty}&fg=000000, we have Poincaré’s recurrence theorem. If {\mu(X)=\infty}&fg=000000, this is not true in general and one has to assume an additional condition, described by the

Definition 1 We say {\mu}&fg=000000 is conservative if {\mu(A)=0}&fg=000000 whenever {A\in\mathcal A}&fg=000000 is such that {\{F^{-n}A\}_{n\ge 0}}&fg=000000 are pairwise disjoint.

Exercise 1 Prove that a conservative measure-preserving system satisfies Poincaré’s recurrence theorem.

Conservativity is also a necessary condition: the translation {x\mapsto x+1}&fg=000000, {x\in\mathbb Z}&fg=000000, is invariant for the counting measure on {\mathbb Z}&fg=000000 but the orbit of every point scapes to infinity. This is indeed, if one assumes ergodicity and invertibility, the only example.

Lemma 2 An invertible ergodic measure-preserving system is non-conservative if and only if it is isomorphic to the translation {x\mapsto x+1}&fg=000000, {x\in\mathbb Z}&fg=000000, with the counting measure on {\mathbb Z}&fg=000000.

Proof: Assume {(X,\mathcal A,\mu,F)}&fg=000000 is an invertible ergodic non-conservative measure-preserving system and let {A\in\mathcal A}&fg=000000 such that {\{F^{-n}A\}_{n\ge 0}}&fg=000000 are pairwise disjoint. Because {F}&fg=000000 is invertible, {\{F^{-n}A\}_{n\in\mathbb Z}}&fg=000000 are pairwise disjoint and thus, by ergodicity,

\displaystyle X=\bigcup_{n\in\mathbb Z}F^{-n}A.&fg=000000

Again by ergodicity, {A}&fg=000000 is a {\mu}&fg=000000-atom (non trivial subsets of {B}&fg=000000 of {A}&fg=000000 give rise to non trivial {F}&fg=000000-invariant sets {\bigcup_{n\in\mathbb Z}F^{-n}B}&fg=000000). This guarantees the map {n\mapsto F^nA}&fg=000000 is an isomorphism from {\mathbb Z}&fg=000000 to {X}&fg=000000. \Box&fg=000000

In the non-invertible situation, there are many other examples. See the end of this paper of J. Aaronson and T. Meyerovitch.

From now on, we always assume {(X,\mathcal A,\mu,F)}&fg=000000 is ergodic and conservative. Let {\phi:X\rightarrow {\mathbb R}}&fg=000000 be a measurable function. A successful area in ergodic theory deals with the convergence of the averages {n^{-1}\cdot\sum_{k=0}^{n-1}\phi\left(F^kx\right)}&fg=000000, {x\in X}&fg=000000, when {n}&fg=000000 goes to infinity. The well known Birkhoff’s theorem states that, if {\mu(X)<\infty}&fg=000000, such limit exists for almost every {x\in X}&fg=000000 whenever {\phi}&fg=000000 is a {L^1}&fg=000000-function. This is not the case when {\mu}&fg=000000 is infinite. Indeed, if {\mu(X)=\infty}&fg=000000, these averages converge to zero for almost every {x\in X}&fg=000000. We give an idea why this is so: if {(X,\mathcal A,\mu,F)}&fg=000000 is a measure-preserving system with {\mu(X)<\infty}&fg=000000, Birkhoff’s theorem asserts that

\displaystyle \dfrac{1}{n}\sum_{k=0}^{n-1}\phi\left(F^kx\right)\longrightarrow\dfrac{1}{\mu(X)}\int_X\phi d\mu\ \ \text { as }n\rightarrow\infty&fg=000000

for {\mu}&fg=000000-almost every {x\in X}&fg=000000. Heuristically, if we take {\mu(X)=\infty}&fg=000000 in the right hand side of the above convergence, then the Birkhoff averages converge to zero. This argument demands a little care, and indeed is more convenient to conclude it as a corollary of Hopf’s ratio ergodic theorem (see below). At this point, it is natural to ask if there exists some “appropriate” rate of convergence: is there a normalizing sequence of constants {(a_n)}&fg=000000 such that {{a_n}^{-1}\cdot\sum_{k=0}^{n-1}\phi\left(F^kx\right)}&fg=000000 converges almost surely? And the answer is no!

Theorem 3 (Aaronson) Let {(X,\mathcal A,\mu,F)}&fg=000000 be a measure-preserving system with {\mu(X)=\infty}&fg=000000, and let {(a_n)}&fg=000000 be a sequence of positive real numbers. Then, for every {\phi\in L^1(X,\mathcal A,\mu)}&fg=000000 such that {\phi\ge 0}&fg=000000 and {\int_X\phi d\mu>0}&fg=000000,

\displaystyle \limsup_{n\rightarrow\infty}\dfrac{\sum_{k=0}^{n-1}\phi\left(F^kx\right)}{a_n}=\infty\ \text{ a.e }\ \ \text{ or }\ \ \liminf_{n\rightarrow\infty}\dfrac{\sum_{k=0}^{n-1}\phi\left(F^kx\right)}{a_n}=0\ \text{ a.e}.&fg=000000

This means that any attempt of normalization will under or overestimate the behavior of Birkhoff sums. Nevertheless, the Birkhoff sums fluctuate in the same proportional rate, according to the following result.

Theorem 4 (Hopf’s ratio ergodic theorem) Let {(X,\mathcal A,\mu,F)}&fg=000000 be a measure-preserving system. Then, for every {\phi,\psi\in L^1(X,\mathcal A,\mu)}&fg=000000 such that {\psi\ge 0}&fg=000000 and {\int_X\psi d\mu>0}&fg=000000,

\displaystyle \dfrac{\sum_{k=0}^{n-1}\phi\left(F^kx\right)}{\sum_{k=0}^{n-1}\psi\left(F^kx\right)}\longrightarrow \dfrac{\int_X\phi\,d\mu}{\int_X\psi d\mu}\ \ \text { as }n\rightarrow\infty&fg=000000

for {\mu}&fg=000000-almost every {x\in X}&fg=000000.

Exercise 2 Prove, using Hopf’s ratio ergodic theorem, that if {(X,\mathcal A,\mu,F)}&fg=000000 is a measure-preserving system with {\mu(X)=\infty}&fg=000000 then, for every {\phi\in L^1(X,\mathcal A,\mu)}&fg=000000,

\displaystyle \dfrac{1}{n}\sum_{k=0}^{n-1}\phi\left(F^kx\right)\longrightarrow 0\ \ \text { as }n\rightarrow\infty&fg=000000

for {\mu}&fg=000000-almost every {x\in X}&fg=000000.

Hopf’s ratio ergodic theorem is an indication that some sort of regularity might exist and it might still be possible, for a specific sequence {(a_n)}&fg=000000, that the averages oscillate without converging to zero or infinity and so one can hope for a summability method that smooths out the fluctuations and forces convergence.

1. Law of large numbers

More generally, one can hope for a law of large numbers.

Definition 5 A law of large numbersfor a measure-preserv\-ing system {(X,\mathcal A,\mu,F)}&fg=000000 is a function {L:\{0,1\}^{\mathbb N}\rightarrow[0,\infty]}&fg=000000 such that, for any {A\in\mathcal A}&fg=000000, the equality

\displaystyle L(\chi_A(x),\chi_A(Fx),\chi_A(F^2x),\ldots)=\mu(A)&fg=000000

holds for {\mu}&fg=000000-almost every {x\in X}&fg=000000.

One can see the function {L}&fg=000000 as a sort of blackbox: given the input of hittings of a generic point {x\in X}&fg=000000 to a fixed set {A\in\mathcal A}&fg=000000, the output is the measure of {A}&fg=000000. For example, if {\mu(X)=1}&fg=000000, the function {L:\{0,1\}^{\mathbb N}\rightarrow[0,\infty]}&fg=000000 defined by

\displaystyle L(x_0,x_1,\ldots)=\left\{ \begin{array}{rl} \displaystyle\lim_{n\rightarrow\infty}\dfrac{1}{n}\sum_{k=0}^{n-1}x_k\ ,&\text{ if the limit exists,}\\ &\\ 0\ ,&\text{ otherwise} \end{array} \right.&fg=000000

is a law of large numbers. The infinite measure situation is quite different: there are systems with no law of large numbers. For example, let {F}&fg=000000 be squashable: there is {G:(X,\mathcal A)\rightarrow(X,\mathcal A)}&fg=000000, commuting with {F}&fg=000000, such that

\displaystyle \mu(G^{-1}A)=c\cdot\mu(A)\ \ \text{for all }A\in\mathcal A, \ \ \ \ \ (1)&fg=000000

for some {c\not=1}&fg=000000. If {F}&fg=000000 had a law of large numbers, say {L}&fg=000000, then for {\mu}&fg=000000-almost every {x\in X}&fg=000000 we would have

\displaystyle \begin{array}{rcl} \mu(A)&=&L(\chi_A(Gx),\chi_A(FGx),\ldots)\\ &&\\ &=&L(\chi_A(Gx),\chi_A(GFx),\ldots)\\ &&\\ &=&L(\chi_{G^{-1}A}(x),\chi_{G^{-1}A}(Fx),\ldots)\\ &&\\ &=&\mu(G^{-1}A), \end{array} &fg=000000

contradicting the assumption of squashability. See An Introduction to Infinite Ergodic Theoryfor more on squashable systems.

There are, fortunately, some conditions that guarantee the existence of law of large numbers.

2. Rational ergodicity

Given {A\in\mathcal A}&fg=000000, let {S_n(A):X\rightarrow{\mathbb N}}&fg=000000 be the Birkhoff sum of the characteristic function {\chi_A}&fg=000000 with respect to {F}&fg=000000.

Definition 6 A measure-preserving system {(X,\mathcal A,\mu,F)}&fg=000000 is called rationally ergodic if there is a sweep-out set {A\in\mathcal A}&fg=000000 with {0<\mu(A)<\infty}&fg=000000 satisfying a Renyi inequality: there is {M>0}&fg=000000 such that

\displaystyle \int_A S_n(A)^2d\mu\ \le \ M\cdot\left(\int_A S_n(A)d\mu\right)^2&fg=000000

for every {n\ge 1}&fg=000000. We say {(X,\mathcal A,\mu,F)}&fg=000000 is rationally ergodic along a subsequence of iterates if the above inequality holds for a subsequence {(n_k)}&fg=000000 of positive integers.

By sweep-out we mean that every point of {X}&fg=000000 eventually enters in {A}&fg=000000, that is

\displaystyle X=\bigcup_{n\ge 0}F^{-n}A.&fg=000000

Definition 7 {(X,\mathcal A,\mu,F)}&fg=000000 is called weakly homogeneous if there is a sequence {(a_{n_k})}&fg=000000 of positive real numbers such that, for all {\phi\in L^1(X,\mathcal A,\mu)}&fg=000000,

\displaystyle \dfrac{1}{N}\sum_{k=1}^N\dfrac{1}{a_{n_k}}\sum_{j=0}^{n_k-1}\phi\left(F^jx\right)\ \longrightarrow\ \int_X\phi d\mu \ \ \text { as }n\rightarrow\infty \ \ \ \ \ (2)&fg=000000

for {\mu}&fg=000000-almost every {x\in X}&fg=000000.

{(a_{n_k})}&fg=000000 is called a return sequence of {F}&fg=000000 and it is unique up to asymptotic equality.

Theorem 8 (Aaronson) Every measure-preserving system {(X,\mathcal A,\mu,F)}&fg=000000 that is rationally ergodic along a subsequence of iterates is weakly homogeneous. More specifically, every subsequence {(a_{n_k})}&fg=000000 can be refined to a further subsequence such that (2) holds for {\mu}&fg=000000-almost every {x\in X}&fg=000000.

Theorem 8 also gives the explicit description of the normalizing constants:

\displaystyle a_{n_k}=\dfrac{1}{\mu(A)^2}\int_A S_{n_k}(A)d\mu =\dfrac{1}{\mu(A)^2}\sum_{j=0}^{n_k-1}\mu\left(A\cap F^{-j}A\right). \ \ \ \ \ (3)&fg=000000

Observe that weak homogeneity defines an explicit law of large numbers {L:\{0,1\}^{\mathbb N}\rightarrow[0,\infty]}&fg=000000 by

\displaystyle L(x_0,x_1,\ldots)=\left\{ \begin{array}{rl} \displaystyle\lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{k=1}^N\dfrac{1}{a_{n_k}}\sum_{j=0}^{n_k-1}x_j\ ,&\text{ if the limit exists,}\\ &\\ 0\ ,&\text{ otherwise.} \end{array} \right.&fg=000000

The next post will focus on a result in collaboration with P. Cirilo and E. Pujals in which we construct a class of cylinder flows that are rationally ergodic along a subsequence of iterates (and thus have explicit law of large numbers).

Previous posts: CF0, CF1, CF2.

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