Geometric rigidity of $\times m$ invariant measures
Geometric rigidity of $\times m$ invariant measures
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DOI：
10.4171/jems/340
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发表时间：
2010-08
影响因子：
2.6
通讯作者：
M. Hochman
中科院分区：
文献类型：
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作者：
M. Hochman
设μ是[0，1]上的概率测度，它对Ta（x）= axmod 1是不变的和遍历的，且0 <dim μ <1。设f是开集上的局部自同态。我们表明，如果E？R和（f µ）|E~µ| E，那么f '（x）？{± a r：r？Q}（µ-a.e.）点x？f-1 E.特别地，如果g是保持μ的分段解析映射，则存在包含supp μ的开g-不变集U，使得g| U是分段线性的，斜率是a的有理幂。类似地，对于上述的μ，如果B是另一个整数，并且a，B不是公共整数的幂，并且如果？是T B不变测度，则f μ？对所有局部的C2类的超同态f。这推广了Rupert-Johnson定理，并表明T a，T B的测度刚性不是阿贝尔作用的结构的结果，而是它们的光滑共轭类的结果：如果U，V是R/Z的映射，它们与T a，T B是C2-共轭的，那么它们就没有正维数的共同测度，而且它们都是遍历的。
Let µ be a probability measure on [0,1] which is invariant and ergodic for T a (x)=axmod1 , and 0<dimµ<1 . Let f be a local diffeomorphism on some open set. We show that if E?R and (fµ)| E ~µ| E , then f ' (x)?{±a r :r?Q} at µ -a.e. point x?f -1 E . In particular, if g is a piecewise-analytic map preserving µ then there is an open g -invariant set U containing supp µ such that g| U is piecewise-linear with slopes which are rational powers of a . In a similar vein, for µ as above, if b is another integer and a,b are not powers of a common integer, and if ? is a T b -invariant measure, then fµ?? for all local diffeomorphisms f of class C 2 . This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of T a ,T b is a result not of the structure of the abelian action, but rather of their smooth conjugacy classes: if U,V are maps of R/Z which are C 2 -conjugate to T a ,T b then they have no common measures of positive dimension which are ergodic for both.