Let $X$ be a Borel probability space (i.e. equipped with a measure $\mu$ on the Borel $\sigma$-algebra such that $\mu(X) = 1$) with a measure-preserving transformation $T$ such that every point has a dense orbit. Does it follow that the measure-preserving system $(X,\mu,T)$ is ergodic?

I have heard that the answer is no, but I haven't been able to think of any examples.

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    $\begingroup$ Furstenberg constructed smooth skew products that are minimal (every orbit is dense), but have uncountably many ergodic measures. A non-trivial linear combination would be an invariant non-ergodic measure. His paper is "Strict ergodicity and transformation of the torus". I recall it also can be done rather easily in symbolic setting. Also, I think, Boshernitzan has a nice construction of a minimal interval exchange (4 intervals?) that is not uniquely ergodic. $\endgroup$ Sep 1, 2011 at 18:48
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    $\begingroup$ Andrey Gogolev also provided an answer, but you want to assume that $X$ is a metric (or at least topological) space, $T$ is continuous and the $\sigma$-algebra is the Borel $\sigma$-algebra. Otherwise the question doesn't really make sense. $\endgroup$ Sep 1, 2011 at 18:57

2 Answers 2


Here is a symbolic construction alluded to by @Andrey Gogolov

Start with two words $W^{(1)}_1=a$ and $W^{(1)}_2=b$.

Then inductively create sequences of pairs of new words by stringing together copies of the words you created at the previous stage. You want to do this in such a way that the dominance of $a$'s in the 1-words and $b$'s in the 2-words does not become diluted.

So for example, try $W^{(n)}_1=[ W^{(n-1)}_1 ] ^ {2^n-1} W^{(n-1)}_2$ and $W^{(n)}_2= [ W^{(n-1)}_2 ]^{2^n-1} W^{(n-1)}_1$ ( so that the $W^{(n)}$ words are $2^n$ times the length of the previous pair of words ). Let $L^{(n)}$ be the length of the blocks at the $n$th stage.

Now let $X$ be the set of all bi-infinite sequences such that each finite block of the sequence occurs in some $W^{(n)}_i$ (and hence all $W^{(m)}_j$ with $m>n$).

Claim that $X$ is minimal (all points have dense orbits): let $B$ be a block appearing in $X$ somewhere. By definition, it's a sub-word of some $W^{(n)}_j$. Now let $x\in X$. Choose a block in $x$ of length exceeding $2L^{(n+1)}$. By definition of $X$ again, this block occurs inside a level $m$ block for some $m>n+1$. Since level $m$ blocks are simply concatenations of level $n+1$ blocks, the original block in $x$ must contain a complete level $n+1$ block. Since the level $n+1$ blocks contain both of the level $n$ blocks, it follows that $x$ contains the block $B$. This establishes the minimality of $X$.

As for lack of unique ergodicity, let $\rho^{(n)}$ denote the density of $a$'s in the level $n$ 1-block (so that $1-\rho^{(n)}$ is the density of $a$'s in the level $n$ 2-block). Recursively, we have $\rho^{(n+1)}=[ (2^{n+1}-1)\rho^{(n)} + (1-\rho^{(n)}) ]/2^{n+1} = (1-2^{-n})\rho^{(n)} + 2^{-(n+1)}$. This is specially chosen to remain bounded away from 1/2. Hence there exists $b>1/2$ such that the density of $a$'s in a level $n$ 1-block is bounded below by $b$, whereas the density in a level $n$ 2-block is bounded above by $1-b$. This contradicts unique ergodicity.

Another very nice family of examples of minimal but non-uniquely ergodic systems was given by Masur and Smillie in the context of polygonal billiards.


While reading Wikipedia on translation surface, a certain kind of flat surface, I tripped across the sentence "there may be directions in which the flow is minimal (meaning every orbit is dense in the surface) but not ergodic" The cited reference is Howard Masur (2006), "Ergodic theory of translation surfaces", Handbook of dynamical systems. Vol. 1B, Handbook of Dynamical Systems, 1, Elsevier B. V., Amsterdam, pp. 527–547, doi:10.1016/S1874-575X(06)80032-9, ISBN 9780444520555, MR 2186247


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