3
$\begingroup$

I think that piecewise expanding maps on the unit interval have been studied. Is there something analogous in dimension 2 with the same good properties? In other words, maybe I want some good results which hold for piecewise expanding maps, and can be extended to higher dimension, such as the results about Perron-Frobenius operators.

$\endgroup$
4
  • $\begingroup$ Expansive maps on $d$-dimensional branched manifolds are considered often in the study of dynamics on 'tiling spaces' and Perron-Frobenius theory is used extensively there. Does that count? $\endgroup$
    – Dan Rust
    Aug 23, 2019 at 15:33
  • $\begingroup$ Is there any reference? $\endgroup$
    – jiaming wu
    Aug 26, 2019 at 9:52
  • $\begingroup$ The starting point is a paper by Anderson and Putnam but there's much more to be said after that, e.g. Sadun's work. $\endgroup$
    – Dan Rust
    Aug 26, 2019 at 12:56
  • $\begingroup$ Thank you . That seems to be an interesting topic $\endgroup$
    – jiaming wu
    Aug 28, 2019 at 2:23

2 Answers 2

1
$\begingroup$

If you are interested in local diffeomorphisms, where the map is continuous everywhere, then Chapters 11 and 12 of the book "Foundations of Ergodic Theory" by Viana and Oliveira has a very good account.

If you want to allow discontinuities, so that the map really is piecewise expanding, then in addition to the paper that Rafael linked to, there is a series of papers by Jerome Buzzi (and some co-authors) from 1997-2003 that would be worth looking at:

$\endgroup$
1
  • $\begingroup$ Thank you for the references. $\endgroup$
    – jiaming wu
    Aug 25, 2019 at 11:16
0
$\begingroup$

Perhaps this paper by P. Eslami will answer your question

https://arxiv.org/pdf/1711.09245.pdf. Check the references as well.

$\endgroup$
1
  • 1
    $\begingroup$ Thank you. I will read the article. $\endgroup$
    – jiaming wu
    Aug 26, 2019 at 9:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.