We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) T T for which the cohomological equation \[ Ψ − Ψ ∘ T = Φ \Psi -\Psi \circ T=\Phi \] has a bounded solution Ψ \Psi provided that the datum Φ \Phi belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The proof is purely dynamical and is based on a renormalization argument and on Gottshalk-Hedlund’s theorem. If the datum is more regular the loss of differentiability in solving the cohomological equation will be the same. The class of interval exchange maps is characterized in terms of a diophantine condition of Roth type imposed to an acceleration of the Rauzy-Veech-Zorich continued fraction expansion associated to T T . More precisely one must impose a growth rate condition for the matrices appearing in the continued fraction algorithm together with a spectral gap condition (which guarantees unique ergodicity) and a coherence condition. We also prove that the set of Roth-type interval exchange maps has full measure. In the appendices we construct concrete examples of Roth-type i.e.m.’s and we show how the growth rate condition alone does not imply unique ergodicity.
A standard interval exchange map is a one-to-one map of the interval that is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type, which is almost surely satisfied in parameter space. Let T0 be a standard interval exchange map of restricted Roth type, and let r be an integer 2. We prove that, amongst C r+3 deformations of T0 that are C r+3 tangent to T0 at the singularities, those that are conjugated to T0 by a C r -diffeomorphism close to the identity form a C 1 -submanifold of codimension (g − 1)(2r + 1) + s.Here, g is the genus and s is the number of marked points of the translation surface obtained by suspension of T0. Both g and s can be computed from the combinatorics of T0.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.