I'm reading Milnor's [notes](http://www.math.stonybrook.edu/%7Ejack/DYNOTES/) on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this example is discussed; no paper by Ricardo Mañé, no books or papers in which this example is mentioned. Milnor also states that an earlier paper by Zakharevich has a similar example, but again no reference is given, and I cannot find this paper anywhere. Has this or any other counterexample been published anywhere else? Any references would be highly appreciated.



Mañé's example of an attractor with no natural measure
**Q:** Has this or any other counterexample been published anywhere else?


**A:** In [A continuous Bowen-Mañé type phenomenon](https://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=3046) examples of vector fields without physical measure for certain parameters are discussed under the name "*Bowen-Mañé type phenomena*". No primary reference to either scientist is given.


The Bowen-Mañé example is also called "*Bowen's eye*" in the literature, see [arXiv: 2010.08945](https://arxiv.org/abs/2010.08945) and [arXiv:1609.05356](https://arxiv.org/abs/1609.05356). The attribution to Bowen is due to Takens, who writes in a [1994 paper:](https://www.emis.de/journals/em/docs/boletim/vol251/v25-1-a6-1994.pdf)



> 
> "I attribute this example to Bowen: although he never published it, I
> learned this example through a paper by Ruelle who referred to Bowen
> (we have no reference: even Ruelle could not localize that paper)."
> 
> 
> 


![](https://i.stack.imgur.com/isY2c.png)
[[source]](https://www.stefanoluzzatto.net/uploads/2/6/7/1/26719559/2017-09_tunisia.pdf)


