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.
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$\begingroup$ 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 RustCommented Aug 23, 2019 at 15:33
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$\begingroup$ Is there any reference? $\endgroup$– jiaming wuCommented Aug 26, 2019 at 9:52
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$\begingroup$ 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 RustCommented Aug 26, 2019 at 12:56
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$\begingroup$ Thank you . That seems to be an interesting topic $\endgroup$– jiaming wuCommented Aug 28, 2019 at 2:23
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2 Answers
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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:
- "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)
answered Aug 23, 2019 at 22:01
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$\begingroup$ Thank you for the references. $\endgroup$ Commented Aug 25, 2019 at 11:16
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Perhaps this paper by P. Eslami will answer your question
https://arxiv.org/pdf/1711.09245.pdf. Check the references as well.
answered Aug 23, 2019 at 16:20
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1$\begingroup$ Thank you. I will read the article. $\endgroup$ Commented Aug 26, 2019 at 9:51