2015/03/31 by Balázs Bárány, BALÁZS BÁRÁNY · 22 citations
Mathematics · Physics and Astronomy · #Analytic and geometric function theory #Chaos control and synchronization #Dimension (graph theory) #Invariant (physics) #Invariant measure #Iterated function #Iterated function system #Lyapunov exponent #Lyapunov function #Mathematical Dynamics and Fractals #Measure (data warehouse) #math.DS #msc:28A80 #msc:37C45
paper · pdf · doi:10.1017/s0305004115000419
published in Mathematical Proceedings of the Cambridge Philosophical Society 159(3), 405-432 (Cambridge University Press)
arxiv created 2015/06/02 · openalex publication_date 2015/08/03 · arxiv updated 2015/11/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier–Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter–Lalley's theorem and we consider self-affine measures and sets generated by lower triangular matrices.