Posted by: matheuscmss | October 27, 2014

A family of maps preserving the measure of ZxT

Last 15th October 2014, the “flat seminar” coorganized by Anton Zorich, Jean-Christophe Yoccoz and myself restarted in a new format: instead of one talk per week, we shifted to one talk per month.

The first talk of this seminar in this new format was given by Alba Málaga, and the next two talks (on next November 12th and December 10th) will be given by Giovanni Forni (on the ergodicity for billiards in irrational polygons) and James Tanis (on equidistribution for horocycle maps): the details can be found here.

In this blog post, we will discuss Alba’s talk about some of the results in her PhD thesis (under the supervision of J.-C. Yoccoz) concerning a family of maps preserving the measure of {\mathbb{Z}\times\mathbb{T}} (as hinted by the title of this post). Of course, any mistakes/errors in what follows are my entire responsibility.

In her PhD thesis, Alba studies the following family of dynamical systems (“cylinder flows”).

The phase space is {\mathbb{Z}\times\mathbb{T}} where {\mathbb{T}=\mathbb{R}/\mathbb{Z}} is the unit circle. We call {\{n\}\times\mathbb{T}} the circle of level {n\in\mathbb{Z}} in the phase space.

The parameter space is {\mathbb{T}^{\mathbb{Z}}:=\{\underline{\alpha}=(\dots, \alpha_{-1}, \alpha_0, \alpha_1,\dots): \alpha_n\in\mathbb{T} \,\,\,\, \forall \, n\in\mathbb{Z}\}}.

Given a parameter {\underline{\alpha}\in \mathbb{T}^{\mathbb{Z}}}, we can define a transformation {F_{\underline{\alpha}}} of the phase space {\mathbb{Z}\times\mathbb{T}} by rotating the elements {(n,x)} of the circle of level {n} by {\alpha_n}, and then by putting them at the level {n+1} (one level up) or {n-1} (one level down) depending on whether they fall in the first or second half of the circle of level {n}. In other terms,

\displaystyle F_{\underline{\alpha}}(n,x):=\left\{\begin{array}{cl} (n+1, R_{\alpha_n}x) & \textrm{if } 0 < R_{\alpha_n}x < 1/2 \\ (n-1, R_{\alpha_n}x) & \textrm{if } -1/2 < R_{\alpha_n}x < 0 \end{array}\right.

where {R_{\alpha}x:=x+\alpha} is the rotation by {\alpha} on the unit circle {\mathbb{T}}.

Note that we have left {F_{\underline{\alpha}}} undefined at the points {(n,x)} such that {R_{\alpha_n}x=0} or {R_{\alpha_n}x=1/2}. Of course, one can complete the definition of {F_{\underline{\alpha}}} by sending each of the points in this countable family to a level up or down in an arbitrary way. However, we prefer not do so because this countable family of points will play no role in our discuss of typical orbits of {F_{\underline{\alpha}}}. Instead, we will think of the set {\textrm{sing}(F_{\underline{\alpha}})} of points where {F_{\underline{\alpha}}} is undefined as a (very mild) singular set.

Alba’s initial motivation for studying this family comes from billiards in irrational polygons. Indeed, our current knowledge of the dynamics of billiard maps on irrational polygons (i.e., polygons whose angles are not all rational multiples of {\pi}) is very poor, and, as Alba explained very well in her talk (with the aid of computer-made figures), she has a good heuristic argument suggesting that the billiard map on an irrational lozenge obtained by small perturbation of an unit square can be thought as a small perturbation of some members of the family {F_{\underline{\alpha}}}. However, we will not pursue further this direction today and we will focus exclusively on the features of {F_{\underline{\alpha}}} from now on.

It is an easy exercise to check that, for any parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}}, the corresponding dynamical system {F_{\underline{\alpha}}} preserves the infinite product measure {\mu=\nu\times \textrm{Leb}}, where {\nu} is the counting measure on {\mathbb{Z}} and {\textrm{Leb}} is the Lebesgue measure on {\mathbb{T}}.

In this setting, Alba’s thesis is concerned with the dynamics of {F_{\underline{\alpha}}} for a typical parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}} (in both Baire-category and measure-theoretical senses).

Before stating some of Alba’s results, let us quickly discuss the dynamical behavior of {F_{\underline{\alpha}}} for some particular choices of the parameter {\underline{\alpha}\in \mathbb{T}^{\mathbb{Z}}}.

Example 1 Consider the constant sequence {(1/2)_{\infty}:=(\dots, 1/2, 1/2, 1/2,\dots)}. By definition, {F_{(1/2)_{\infty}}} acts by a translation by {1/2} on the {x}-coordinate of all points {(n,x)} of the phase space. In particular, the second iterate {F_{(1/2)_{\infty}}^2(n,x)} of any point {(n,x)} has the form {F_{(1/2)_{\infty}}^2(n,x) = (c_{1/2}(n,x),x)} where {c_{1/2}(n,x)\in\mathbb{Z}}. Furthermore, the function {c_{1/2}(n,x)} is not difficult to compute: since {0<x<1/2 \iff 1/2<x+1/2<1}, we see that if {0<x<1/2}, resp. {1/2<x<1}, then {F_{(1/2)_{\infty}}(n,x) = (n-1,x+1/2)} resp. {(n+1,x+1/2)} and, hence, {F_{(1/2)_{\infty}}^2(n,x)=(n,x)}. In other words, {c_{1/2}(n,x)=n} for all {(n,x)}, and, thus, {F_{(1/2)_{\infty}}} is a periodic transformation (of period two).

Example 2 Consider the constant sequence {(1/3)_{\infty}:=(\dots, 1/3, 1/3, 1/3, \dots)}. Similarly to the previous example, {F_{(1/3)_{\infty}}} acts periodically (with period {3}) on the {x}-coordinate in the sense that {F_{(1/3)_{\infty}}^3(n,x)=(c_{1/3}(n,x),x)} where {c_{1/3}(n,x)\in\mathbb{Z}}. Again, the function {c_{1/3}(n,x)} is not difficult to compute: by dividing the unit circle {\mathbb{T}} into the six intervals {I_j = \left[ \frac{j}{6}, \frac{j+1}{6} \right]}, {j=0,\dots, 5}, one can easily check that

\displaystyle c_{1/3}(n,x)=\left\{\begin{array}{cl} n+1 & \textrm{if } x\in I_j \textrm{ with } j \textrm{ even } \\ n-1 & \textrm{if } x\in I_j \textrm{ with } j \textrm{ odd } \end{array} \right.

In particular, we see that {F_{(1/3)_{\infty}}^3} systematically moves the copy {\{n\}\times I_j} of an interval {I_j} with {j} even, resp. odd, at the circle of level {n} to the corresponding copy {\{n+1\}\times I_j}, resp. {\{n-1\}\times I_j}, of the interval {I_j} at level {n+1}, resp. {n-1}. In other terms, {F_{(1/3)_{\infty}}} has wandering domains (i.e., domains which are disjoint from all its non-trivial iterates under the map) of positive {\mu}-measure and, hence, {F_{(1/3)_{\infty}}} is not conservative in the sense that it does not satisfy Poincaré’s recurrence theorem with respect to the infinite invariant measure {\mu}: for example, for each {k\in\mathbb{N}}, {F_{(1/3)_{\infty}}^{3k}} sends the subset {\{0\}\times I_0} of {\mu}-measure {1/6} always “upstairs” to its copy {\{k\}\times I_0} at the {k}-th level, so that the orbits of points in {\{0\}\times I_0} escape to {+\infty} (one of the “ends”) in the phase space {\mathbb{Z}\times\mathbb{T}}.

Remark 1 The reader can easily generalize the previous two examples to obtain that the transformation {F_{(p/q)_{\infty}}} associated to the constant sequence {(p/q)_{\infty} = (\dots, p/q, p/q, p/q,\dots)} with {p/q\in\mathbb{Q}} (a rational number written in lowest terms) is periodic or it has wandering domains of positive measure depending on whether the denominator {q\in\mathbb{N}} is even or odd.

Example 3 By a theorem of Conze and Keane, the transformation {F_{(\alpha)_{\infty}}} associated to a constant sequence {(\alpha)_{\infty} = (\dots, \alpha, \alpha, \alpha, \dots)} with {\alpha\in\mathbb{R}-\mathbb{Q}} is ergodic (but not minimal).

Today, we will give sketches of the proofs of the following two results:

Theorem 1 (Málaga) For almost all parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}} (with respect to the standard product Lebesgue measure), the transformation {F_{\underline{\alpha}}} is conservative, i.e., {F_{\underline{\alpha}}} has no wandering domains of positive {\mu}-measure.

Theorem 2 (Málaga) For a Baire-generic parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}} (with respect to the standard product topology), the transformation {F_{\underline{\alpha}}} is conservative, ergodic, and minimal.

1. Conservativity of {F_{\underline{\alpha}}} for typical parameters

Let {\underline{\alpha}=(\dots, \alpha_{-1}, \alpha_0, \alpha_1,\dots)\in\mathbb{T}^{\mathbb{Z}}}. We say that the circle of level {n} is mirror for {F_{\underline{\alpha}}} if {\alpha_n=1/2}. This nomenclature is justified by the fact that any orbit of {F_{\underline{\alpha}}} hitting a mirror level {n} ends up by bouncing back:

  • by definition, a point {(n-1,x)} sent upstairs by {F_{\underline{\alpha}}} to the point {F_{\underline{\alpha}}(n-1,x)=(n, R_{\alpha_{n-1}}x)} at the level {n} satisfies {0<R_{\alpha_{n-1}}x<1/2}; thus, the fact that {\alpha_n=1/2} at a mirror level {n} forces the next iterate to come back to the level {n-1}, i.e.,

    \displaystyle F_{\underline{\alpha}}^2(n-1,x) = F_{\underline{\alpha}}(n, R_{\alpha_n-1}x) = (n-1, R_{\alpha_{n-1}+1/2}x)

    because {0<R_{\alpha_{n-1}}x<1/2} if and only if {1/2<R_{\alpha_{n-1}+1/2}x = R_{\alpha_{n-1}}x+1/2<1}; in other terms, a point sent upstairs by {F_{\underline{\alpha}}} to a mirror level is reflected downstairs to the initial level in the next iteration;

  • similarly, a point {(n+1,x)} sent downstairs by {F_{\underline{\alpha}}} to a mirror level {n} is reflected upstairs to the level {n+1} in the next iteration.

Inspired by the notion of mirror levels, let us introduce the set of parameters:

\displaystyle Cons=\{\underline{\alpha} = (\dots, \alpha_{-1}, \alpha_0, \alpha_1, \dots)\in\mathbb{T}^{\mathbb{Z}}: \liminf\limits_{n\rightarrow\pm\infty} d_{\mathbb{T}}(\alpha_n, 1/2) = 0 \}

The next proposition follows directly from the definitions of the standard product topology and Lebesgue measure on {\mathbb{T}^{\mathbb{Z}}}, and it is left as en exercise to the reader:

Proposition 3 {Cons} is a dense {G_{\delta}} subset of {\mathbb{T}^{\mathbb{Z}}} with full measure.

Here, we recall that a {G_{\delta}} subset is a subset containing a countable intersection of open subsets, and, by definition, we say that a given property holds for a Baire-generic parameter {\alpha\in\mathbb{T}^{\mathbb{Z}}} whenever this property is verified for all parameters in a certain {G_{\delta}} dense subset.

In particular, Proposition 3 implies that the first part of Theorem 2 and the entire statement of Theorem 1 are both immediate consequences of the following result:

Proposition 4 If {\underline{\alpha}\in Cons}, then {F_{\underline{\alpha}}} is conservative.

Let us now complete our outline of the proof of Theorem 1 (and the first part of Theorem 2) by giving the basic idea behind of the proof of Proposition 4 (while hiding the details under the rug).

The main point is the following simple variant of the notion of mirror level. Suppose that {n\in\mathbb{Z}} is a level such that {\alpha_n} is very close to {1/2}. Of course, the level {n} is not a (perfect) mirror if {\alpha_n\neq 1/2}, but a direct computation reveals that it is an almost mirror: the set of points passing through the level {n} under iteration by {F_{\underline{\alpha}}} (i.e., not bouncing back to the previous level) has measure {2|\alpha_n-1/2|} because this set is the union of two intervals of sizes {|\alpha_n-1/2|} (containing {1/2} in their boundaries) at the circles of levels {n-1} and {n+1}.

Using the notion of almost mirror levels, a rough outline of the proof of Proposition 4 goes as follows. By contradiction, assume that {F_{\underline{\alpha}}} is not conservative, i.e., there exists a (wandering) set {A} of positive {\mu}-measure whose iterates under {F_{\underline{\alpha}}} are mutually disjoint. By the Lebesgue density theorem, we can essentially think of {A} as an interval of positive Lebesgue measure (around a density point of {A}) at a circle of fixed level (say {0}).

Note that the iterates of {A} escape to infinite (by going to {+\infty} or {-\infty} on the phase space {\mathbb{Z}\times\mathbb{T}}): this happens because the iterates of {A} are all mutually disjoint and their {\mu}-measures are equal to {\mu(A)>0} (since {F_{\underline{\alpha}}} preserves the measure {\mu}), and the {\mu}-measure of any “box” {B_N} consisting of the union of circles of level {n} with {|n|<N} is finite for all {N\in\mathbb{N}}. So, the number {K(N)} of iterates of {A} trapped inside a given box {B_N} is finite since it verifies

\displaystyle K(N)\cdot\mu(A)=\mu(\bigcup\limits_{F^j(A)\subset B_N} F^j(A))\leq \mu(B_N)=2N<\infty.

On the other hand, by definition of {\underline{\alpha}\in Cons}, given {\varepsilon>0}, there are arbitrarily large numbers {N(\varepsilon),M(\varepsilon)\in\mathbb{N}} such that {|\alpha_{N(\varepsilon)}-1/2|<\varepsilon} and {|\alpha_{-M(\varepsilon)}-1/2|<\varepsilon}, i.e., the levels {N(\varepsilon)} and {-M(\varepsilon)} are {\varepsilon}-almost mirrors.

In particular, since {\varepsilon>0} can be chosen arbitrarily small, the set of points whose {F_{\underline{\alpha}}} iterate is not reflected back by a {\varepsilon}-almost mirror has tiny {\mu}-measure (equal to {2\varepsilon}) and {F_{\underline{\alpha}}} preserves the measure {\mu}, one can show that the {\mu}-measure of {A} would be equal arbitrarily small, i.e., {\mu(A)=0}: this occurs because the {F_{\underline{\alpha}}}-iterates of the wandering domain {A} go to either {+\infty} or {-\infty} on the phase space {\mathbb{Z}\times\mathbb{T}}, and, in their way to infinite, they will pass through all {\varepsilon}-almost mirrors with {\varepsilon>0} arbitrarily small located at either the levels {N(\varepsilon)\in\mathbb{N}} or the levels {-M(\varepsilon)\in\mathbb{Z}-\mathbb{N}}. Of course, this is a contradiction with our assumption that {A} is a wandering domain of {F_{\underline{\alpha}}} with {\mu(A)>0}, so that our brief sketch of proof of Proposition 4 is complete.

2. Ergodicity of {F_{\underline{\alpha}}} for (Baire) generic parameters

Recall from the previous section that Theorem 1 and the first part of Theorem 2 were direct consequences of Propositions 3 and 4. Thus, it remains only to show that {F_{\underline{\alpha}}} is ergodic and minimal for a Baire-generic parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}}.

Since the argument to show the minimality of {F_{\underline{\alpha}}} is very similar to the one proving the ergodicity of {F_{\underline{\alpha}}} (both arguments are based on results of minimality and unique ergodicity for interval exchange transformations and translation flows on translation surfaces; cf. Remark 2), from now on we will focus exclusively on the ergodicity of {F_{\underline{\alpha}}} for Baire-generic paramaters {\underline{\alpha}}.

By Proposition 3, our task is reduced to show that

Proposition 5 {Erg=\{\underline{\alpha}\in Cons: F_{\underline{\alpha}} \textrm{ is ergodic}\}} is a dense {G_{\delta}}-subset of {Cons}.

The fact that the condition “{F_{\underline{\alpha}}} is ergodic” leads to a {G_{\delta}} subset is almost due to Oxtoby-Ulam: indeed, Oxtoby-Ulam observed that the ergodicity condition (written in terms of Birkhoff averages) usually leads to {G_{\delta}} sets in the setting of probability measure preserving transformations; of course, {F_{\underline{\alpha}}} preserves an infinite measure, but, as we shall see in a moment, Oxtoby-Ulam’s argument can be adapted this context.

Let {\underline{\alpha}\in Cons}, so that {F_{\underline{\alpha}}} is conservative. In this case, we have a well-defined countable family of first-return maps {R_n} of the orbits of {F_{\underline{\alpha}}} to the circle {\{n\}\times\mathbb{T}} of level {n\in\mathbb{Z}}. Note that each {R_n} preserves the natural Lebesgue (probability) measure on {\{n\}\times\mathbb{T}}.

We affirm that the ergodicity of {F_{\underline{\alpha}}} is detected by the countable family {R_n} of probability measure preserving transformations, i.e., {F_{\underline{\alpha}}} is ergodic if and only if {R_n} is ergodic for all {n\in\mathbb{Z}}. In fact, if {F_{\underline{\alpha}}} is ergodic, then each {R_n} must be ergodic (otherwise the {F_{\underline{\alpha}}}-iterates of a non-trivial {R_n}-invariant subset of {\{n\}\times\mathbb{T}} would give a subset of {\mathbb{Z}\times\mathbb{T}} contradicting the ergodicity of {F_{\underline{\alpha}}}), and, conversely, if {R_n} is ergodic for all {n\in\mathbb{Z}}, then a {F_{\underline{\alpha}}}-invariant with positive measure must be trivial (as it intersects a circle of some level {m} in a set of positive measure, the ergodicity of {R_m} implies that actually it intersects the circle of level {m} in a subset of full measure; hence, by iterating under {F_{\underline{\alpha}}} and using the ergodicity of {R_n} for all {n\in\mathbb{Z}}, we conclude that it intersects all circles of all levels in subsets of full measures).

Now, we will combine this claim together with Oxtoby-Ulam’s argument to show that the set {Erg} is a {G_{\delta}} subset. For this sake, we select a dense subset {\{\varphi_i\}_{i\in\mathbb{N}}\subset L^2(\mathbb{T})} of continuous functions on {\mathbb{T}}, and we observe that the claim above (and the definition of ergodicity in terms of Birkhoff averages) implies that {\underline{\alpha}\in Erg} if and only if {\underline{\alpha}\in\bigcap\limits_{i\in\mathbb{N}} \bigcap\limits_{n\in\mathbb{Z}} \bigcap\limits_{k\in\mathbb{N}} W(\varphi_i, R_n, 1/k)} where

\displaystyle \begin{array}{rcl} W(\varphi_i, R_n, 1/k) &=& \bigcup\limits_{m\in\mathbb{N}}\left\{ \underline{\alpha}\in Cons: \left\|\frac{1}{m}\sum\limits_{l=0}^{m-1}\varphi_i \circ R_n^l - \int \varphi_i \right\|_{L^2} < \frac{1}{k}\right\} \\ &:=& \bigcup\limits_{m\in\mathbb{N}} W_m(\varphi_i, R_n, 1/k) \end{array}

(and, by abuse of notation, we think of the function {\varphi_i} as defined on {\{n\}\times\mathbb{T}} when writing {\varphi_i\circ R_n^l}). Since the parameter sets {W_m(\varphi_i, R_n, 1/k)}, and, a fortiori, {W(\varphi_i, R_n, 1/k)}, can be shown to be relatively open in {Cons} for each fixed {i\in\mathbb{N}}, {n\in\mathbb{Z}}, {k\in\mathbb{N}} and {m\in\mathbb{N}} (the details are left as an exercise to the reader), we deduce that {Erg} is a {G_{\delta}}-subset.

At this point, it remains only to show that {Erg} is a dense subset of {Cons} in order to complete the proof of Proposition 5. By Baire’s theorem, it suffices to prove that {W(\varphi_i, R_n, 1/k)} is a dense subset for each fixed {i, n, k}.

Given {\underline{\alpha}^{(0)} = (\dots, \alpha^{(0)}_{-1}, \alpha^{(0)}_0, \alpha^{(0)}_1,\dots)\in Cons} and a neighborhood {V} of {\underline{\alpha}^{(0)}}, we want to find {\underline{\alpha}\in V\cap W(\varphi_i, R_n, 1/k)}.

Because the basis of neighborhoods of {\underline{\alpha}^{(0)}} in the standard product topology of {\mathbb{T}^{\mathbb{Z}}} is generated by open sets of the form

\displaystyle \{\underline{\alpha}: |\alpha_l - \alpha_l^{(0)}|<\varepsilon \textrm{ for all } l\in E\}

where {\varepsilon>0} and {E\subset \mathbb{N}} is a finite set, we can take {N\in\mathbb{N}} large enough so that {|n|<N} and {V} contains an element {\underline{\beta}} with {\beta_{N}=\beta_{-N}=1/2}.

By definition, the circle at the levels {N} and {-N} are mirrors for {F_{\underline{\beta}}}, so that {F_{\underline{\beta}}} has part of dynamics confined into the box {B_N=\bigcup\limits_{|n|\leq N} \{n\}\times\mathbb{T}}. Furthermore, the reader can verify that the restriction of {F_{\underline{\beta}}} can be interpreted as an interval exchange transformation related to the vertical translation flow on a (compact) translation surface {P_{\underline{\beta}}} obtained by gluing the pieces of the boundaries of the cylinders {[n, n+1]\times\mathbb{T} \subset \mathbb{R}\times\mathbb{T}} accordingly to the formulas defining {F_{\underline{\alpha}}}. Moreover, a “translation”

\displaystyle (\underline{\beta}+t)_a:=\left\{\begin{array}{rl} \beta_a + t & \textrm{ if } |a|\leq N \\ \beta_a & \textrm{ otherwise }\end{array}\right.

of the parameter {\underline{\beta}} by a small quantity {t\in\mathbb{R}} leads to a transformation {F_{\underline{\beta}+t}|_{B_N}} associated to the translation flow on {P_{\underline{\beta}}} is an almost vertical direction.

By a theorem of Kerckhoff, Masur and Smillie, the translation flow on the compact translation surface {P_{\underline{\beta}}} is uniquely ergodic in almost all directions. Using this result, we can choose {t\in\mathbb{R}} so small that {\underline{\alpha} = \underline{\beta}+t\in V} is a parameter such that {F_{\underline{\alpha}}|_{B_N}} is uniquely ergodic. From this fact, it follows that {\underline{\alpha}\in V} also belongs to {W(\varphi_i, R_n, 1/k)} (and, actually, {\bigcap\limits_{i, k\in\mathbb{N}} W(\varphi_i, R_n, 1/k)}), as desired.

Remark 2 The same argument above works by replacing “uniquely ergodic” by “minimal”.

Remark 3 A “counter-intuitive” feature of the previous argument (which is somewhat common in Baire genericity type arguments) is that the denseness of {Erg} (and its analog consisting of minimal dynamics parameters) actually uses non-ergodic (and non-minimal) transformations {F_{\underline{\alpha}}} whose dynamics has a piece blocked inside the box {B_N} between the mirror levels {\pm N}. Of course, the main point is that {F_{\underline{\alpha}}} is “ergodic (and minimal) on a large portion” of the phase space, and this kind of “partial information” is usually sufficient to run a denseness proof based on Baire’s theorem.

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