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The cohomological equation for Roth-type interval exchange maps. (English) Zbl 1112.37002
An interval exchange map \(T\) is said to be of Roth type if it satisfies Keane’s condition (and so has every orbit dense) and three further properties: (a) a growth rate condition concerning the matrices that appear in an accelerated Rauzy-Veech-Zorich continued fraction algorithm, (b) a spectral gap condition, which ensures that \(T\) is uniquely ergodic, and (c) a coherence condition that is used to ensure that certain infinite sums converge.
Two main results are presented. First, if \(T\) is a Roth-type interval exchange map defined on a collection of intervals \(I_\alpha\) and \(\phi\) is a map whose derivative on each \(I_\alpha\) has bounded variation and satisfies another technical condition, then there is a function \(\chi\), constant on each \(I_\alpha\) and a bounded solution \(\psi\) to the cohomological equation \(\psi - \psi\circ T = \phi - \chi\). (By replacing \(\phi\) with a different antiderivative of \(\phi'\) on each \(I_\alpha\), \(\chi\) can be taken to be 0.) The proof is based upon renormalization and a variant of the theorem of Gottschalk and Hedlund that there is a continuous function \(\beta\) satisfying \(\beta\circ f - \beta = \gamma\) whenever \(f\) is a minimal homeomorphism and \(\gamma\) is a continuous function whose Birkhoff sums \(\sum_{j=0}^n\gamma(f^j(p))\) for some point \(p\) are bounded. It is also shown that greater regularity in the data leads to greater regularity in \(\psi\) (essentially, if \(\phi\) is \(C^r\), then \(\psi\) is \(C^{r-2}\) and \(D^{r-2}\psi\) is Lipschitz).
The second main result is that the class of Roth-type interval exchange maps has full measure in the space of all interval exchange maps. Two appendices contain an explicit construction of some Roth-type interval exchanges, and an example showing that condition (a) does not imply unique ergodicity, from which it follows that (a) does not imply (b).

MSC:
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
11K50 Metric theory of continued fractions
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
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