I'm reading Milnor's notes 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.
1 Answer
$\begingroup$
$\endgroup$
2
Q: Has this or any other counterexample been published anywhere else?
A: In A continuous Bowen-Mañé type phenomenon 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 and arXiv:1609.05356. The attribution to Bowen is due to Takens, who writes in a 1994 paper:
"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)."
answered Mar 2, 2022 at 15:00
-
$\begingroup$ Do you think "localize" = "locate" here? $\endgroup$ Mar 2, 2022 at 15:24
-
$\begingroup$ yes, that is certainly what Takens meant. $\endgroup$ Mar 2, 2022 at 15:28