Hi all,

I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and Bernoulli convolutions satisfy something which I call fibre mixing. I'm interested in whether this notion of fibre mixing is something which is more generally studied or understood in the ergodic theory literature.

We begin with a very simple example.

Let $T:[0,1]^2\to[0,1]^2$ be a hyperbolic toral automorphism, perhaps $T(x,y)=(x+2y,x+y)$ mod 1. T preserves two dimensional Lebesgue measure $\lambda^2$ and is mixing, which means that for any Borel sets $A,B\subset[0,1]^2$ we have that $$ \lim_{n\to\infty}\lambda^2(A\cap T^{-n}(B))\to\lambda^2(A)\lambda^2(B). $$ This definition of mixing tells us nothing about sets of zero measure.

However in this situation we can also pass to a lower dimensional form of mixing. Let $A_x:=\lbrace(x',y')\in[0,1]^2:x'=x\rbrace$. Then for each $x\in[0,1]$ we have have that for every Borel subset A of $A_x$, $$ \lim_{n\to\infty}\lambda(A\cap T^{-n}(B))\to\lambda(A)\lambda^2(B), $$ where $\lambda$ is one dimensional Lebesgue measure (and, in particular, $\lambda$ is the conditional measure on $A_x$ induced by $\lambda^2$ on $[0,1]^2$). The proof is not long. This second statement allows us to say something about sets of zero 2-dimensional Lebesgue measure, which our original statement of mixing does not.

As a more interesting example, let $\Sigma=\lbrace0,1\rbrace^2$ and $\beta\in(1,2)$. For $x\in[0,\frac{1}{\beta-1}]$ we let $$\mathcal E_{\beta}(x):=\lbrace\underline a\in\Sigma:\sum_{i=1}^{\infty}a_i\beta^{-i}=x\rbrace.$$ Let m be the $(1/2,1/2)$ Bernoulli measure on $\Sigma$ and $m_x$ be the conditional measure on the fibre $\mathcal E_{\beta,x}$. The shift map $\sigma$ on $(\Sigma,m)$ is mixing. A highly non-trivial theorem of Lindenstrauss, Peres and Schlag says that, for almost every $\beta\in(1,2)$ and for almost every $x\in [0,\frac{1}{\beta-1}]$ we have that for each cylinder set $B\subset \Sigma$ and $m_x$-measurable $A\subset\mathcal E_{\beta}(x)$ we have that
$$ \lim_{n\to\infty} m_x(A\cap T^{-n}(B))\to m_x(A)m(B). $$

The almost everywhere in this statement is strict, the statement fails if $\beta$ is a Pisot number.

Since I don't know of any better terminology, I call each of these mixing statements which involve lower dimensional measures 'fibre mixing results'. For me a fibre mixing result is one where one can condition a measure in a dynamical system on fibres and then replace some occurrences of the original measure in the definition of mixing with the conditioned measure on a fibre.

My questions are: Does this notion already exist in the literature anywhere? Has it been studied? Are there any other non-trivial examples for which one knows that conditioning of a measure leads to fibre mixing?

Any references to anything related would be really helpful.

Thanks a lot, Tom

  • 1
    $\begingroup$ There is a paper by Dan Rudolph titled "Pointwise and L^1 mixing relative to a sub-sigma algebra." His notion of pointwise-relatively mixing seems related. $\endgroup$ – Robin Tucker-Drob Jan 21 '13 at 17:28
  • $\begingroup$ Thanks Robin, this does indeed seem related. I find passing to sub sigma algebras difficult, so I can't at first glance tell whether this is exactly the thing I'm looking for, but in any case this will make me sit down and work out the relationship between sub sigma algebras and factors which will be very good for me! $\endgroup$ – Tom Kempton Jan 23 '13 at 8:29

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