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 PerronFrobenius operators.

$\begingroup$ Expansive maps on $d$dimensional branched manifolds are considered often in the study of dynamics on 'tiling spaces' and PerronFrobenius theory is used extensively there. Does that count? $\endgroup$– Dan RustAug 23, 2019 at 15:33

$\begingroup$ Is there any reference? $\endgroup$– jiaming wuAug 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 RustAug 26, 2019 at 12:56

$\begingroup$ Thank you . That seems to be an interesting topic $\endgroup$– jiaming wuAug 28, 2019 at 2:23
2 Answers
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 coauthors) from 19972003 that would be worth looking at:
 "Intrinsic ergodicity of affine maps in $[0,1]^d$", Monatsh. Math., 1997
 "Markov extensions for multidimensional dynamical systems", Israel J. Math, 1999
 "Absolutely continuous invariant measures for arbitrary expanding piecewise Ranalytic mappings of the plan", ETDS, 2000
 "Conformal measures for multidimensional piecewise invertible maps", ETDS, 2001 (with Paccaut and Schmitt)
 "Thermodynamical formalism for piecewise invertible maps: absolutely continuous invariant measures as equilibrium states, 2001
 "Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps", ETDS, 2003 (with Sarig)
Perhaps this paper by P. Eslami will answer your question
https://arxiv.org/pdf/1711.09245.pdf. Check the references as well.

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