Piecewise expanding map 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.
 A: 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:


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*"Intrinsic ergodicity of affine maps in $[0,1]^d$", Monatsh. Math., 1997

*"Markov extensions for multi-dimensional dynamical systems", Israel J. Math, 1999

*"Absolutely continuous invariant measures for arbitrary expanding piecewise R-analytic 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)

A: Perhaps this paper by P. Eslami will answer your question
https://arxiv.org/pdf/1711.09245.pdf. Check the references as well.
