2016/04/13 by Kalinin, Boris, Sadovskaya, Victoria · 2 citations
#37D #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1604.03963
Let f be a measure-preserving transformation of a Lebesgue space (X,μ) and let \f be its extension to a bundle \E = X ×\Rm by smooth fiber maps \fx : \Ex → \Efx so that the derivative of \f at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes \hx on \Ex for μ-a.e. x so that the maps \px =\hfx ∘ \fx ∘ \hx -1 are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such \hx and \px are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change \h also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism f with a non-uniformly contracting invariant foliation W. We construct a measurable system of smooth coordinate changes \hx: Wx → TxW such that the maps \hfx ∘ f ∘ \hx -1 are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.