Let \(\nu \) be a probability measure that is ergodic under the endomorphism \((\times p, \times p)\) of the torus \({\mathbb {T}}^2\), such that \(\dim \pi \mu < \dim \mu \) for some non-principal projection \(\pi \). We show that, if both \(m\ne n\) are independent of p, the \((\times m, \times n)\) orbits of \(\nu \) typical points will equidistribute towards the Lebesgue measure. If \(m>p\) then typically the \((\times m, \times p)\) orbits will equidistribute towards the product of the Lebesgue measure with the marginal of \(\mu \) on the y-axis. We also prove results in the same spirit for certain self similar measures. These are higher dimensional analogues of results due (among others) to Host, Lindenstrauss, and Hochman-Shmerkin.

设\(\nu \)是在环面\({\mathbb {T}}^2\)的内同态\((\times p, \times p)\)下遍历的概率测度，使得\ (\dim \pi \mu < \dim \mu \)对于一些非主要投影\(\pi \)。我们表明，如果\(m\ne n\)都与p无关，则\(\nu \)典型点的\((\times m, \times n)\)轨道将向 Lebesgue 测度均匀分布。如果\(m>p\)那么通常\((\times m, \times p)\)轨道将向 Lebesgue 测度的乘积等分布，边际为\(\mu \)在y轴上。我们还以相同的精神证明了某些自我相似的措施的结果。这些是由 Host、Lindenstrauss 和 Hochman-Shmerkin 产生的结果的更高维度类似物（除其他外）。