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There is a staggering variety of measures invariant under $\times d$; one cannot expect to describe them all in any explicit way. The Gibbs measures described by Vaughn Climenhaga are very special (they have very strong statistical properties which general invariant measures lack), although of course they are very important (they arise naturally for example when trying to compute Hausdorff dimension).

The support of a $\times d$-invariant measure is a compact $\times d$-invariant set; you cannot say anything more than this (conversely, any compact $\times d$-invariant set supports one, and in fact lots, of $d$-invariant measures).

Even though one cannot give any sort of explicit description of a $\times d$-invariant and ergodic measure, there are many properties these measures have which are not shared by general measures. For example, if $\mu$ is $\times d$-invariant and ergodic, then there is $\alpha\in [0,1]$ such that $$\lim_{r\to 0} \frac{\log(\mu(B(x,r)))}{\log r} = \alpha$$ for $\mu$-almost every $x$ (in particular this is saying the limit on the LHS exists). This is essentially the Shannon-McMillan-Breiman Theorem. Moreover, $\alpha=1$ if and only if $\mu$ is Lebesgue measure.

For $d=2$ you cannot do a straightforward Cantor construction, but you can look at the set of all points in $[0,1]$ such that the binary expansion does not have two consecutive zeros (for example). This is a topological Cantor set of dimension less than $1$. There are many invariant measures supported on the set (such as Markov measures). This is in some sense the simplest nontrivial $\times 2$-invariant set.

Alternatively, you can first do a Cantor construction for $d=4$ (call the resulting measure $\mu$) and then get a $\times 2$-invariant measure out of it by setting $\nu = \frac{1}{2}(\mu + T_2\mu)$ where $T_2=\times 2$.

Although not directly related to your question, let me recall one of the most famous open problems in ergodic theory. There are lots of measures invariant and ergodic under each of $\times 2$ and $\times 3$. Lebesgue measure, as well as some discrete measures, are invariant under both $\times 2$ and $\times 3$. Are there any others? (The answer is known to be "no" if the $\alpha$ in the displayed equation above is not zero.)

1

There is a staggering variety of measures invariant under $\times d$; one cannot expect to describe them all in any explicit way. The Gibbs measures described by Vaughn Climenhaga are very special (they have very strong statistical properties which general invariant measures lack), although of course they are very important (they arise naturally for example when trying to compute Hausdorff dimension).

The support of a $\times d$-invariant measure is a $\times d$-invariant set; you cannot say anything more than this (conversely, any $\times d$-invariant set supports one, and in fact lots, of $d$-invariant measures).

Even though one cannot give any sort of explicit description of a $\times d$-invariant and ergodic measure, there are many properties these measures have which are not shared by general measures. For example, if $\mu$ is $\times d$-invariant and ergodic, then there is $\alpha\in [0,1]$ such that $$\lim_{r\to 0} \frac{\log(\mu(B(x,r)))}{\log r} = \alpha$$ for $\mu$-almost every $x$ (in particular this is saying the limit on the LHS exists). This is essentially the Shannon-McMillan-Breiman Theorem. Moreover, $\alpha=1$ if and only if $\mu$ is Lebesgue measure.

For $d=2$ you cannot do a straightforward Cantor construction, but you can look at the set of all points in $[0,1]$ such that the binary expansion does not have two consecutive zeros (for example). This is a topological Cantor set of dimension less than $1$. There are many invariant measures supported on the set (such as Markov measures). This is in some sense the simplest nontrivial $\times 2$-invariant set.

Alternatively, you can first do a Cantor construction for $d=4$ (call the resulting measure $\mu$) and then get a $\times 2$-invariant measure out of it by setting $\nu = \frac{1}{2}(\mu + T_2\mu)$ where $T_2=\times 2$.

Although not directly related to your question, let me recall one of the most famous open problems in ergodic theory. There are lots of measures invariant and ergodic under each of $\times 2$ and $\times 3$. Lebesgue measure, as well as some discrete measures, are invariant under both $\times 2$ and $\times 3$. Are there any others? (The answer is known to be "no" if the $\alpha$ in the displayed equation above is not zero.)