2016/10/31 by Gerhard Keller, Atsuya Otani · 1 citation
Mathematics · Physics and Astronomy · #Attractor #Base (topology) #Bounded function #Chaos control and synchronization #Dynamical systems theory #Hausdorff dimension #Invariant (physics) #Julia set #Mathematical Dynamics and Fractals #Measure (data warehouse) #Quantum chaos and dynamical systems #Schwarzian derivative #Skew #math.DS #msc:27D25 #msc:27D30 #msc:27D35
paper · pdf · doi:10.1142/s0219493718500090
published as Stochastics & Dynamics 18 (2) (2017) · 8 figures
arxiv created 2016/10/31 · openalex created_date 2016/11/04 · openalex publication_date 2017/03/16 · arxiv updated 2018/03/01 · openalex updated_date 2026/08/05
We consider skew product dynamical systems [Formula: see text] with a (generalized) baker transformation [Formula: see text] at the base and uniformly bounded increasing [Formula: see text] fibre maps [Formula: see text] with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for [Formula: see text], we prove that the presence of these fibres restricts considerably the possible structures of invariant measures — both topologically and measure theoretically, and that this finally allows to provide a “thermodynamic formula” for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.