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
