2015/07/25 by Ratna Pal, Pal, Ratna · 1 citation
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #math.DS
paper · pdf · doi:10.48550/arxiv.1507.07097
83 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.03431; text overlap with arXiv:0810.0811 by other authors
arxiv created 2015/07/25 · openalex publication_date 2015/07/25 · arxiv updated 2015/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the first part of the thesis, we study some dynamical properties of skew products of Hénon maps of \mbb C2 that are fibered over a compact metric space M. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. Further, it is shown that the successive pullbacks of a suitable current by the skew Hénon maps converge to a multiple of the fibered stable current. Second part of the thesis generalizes most of the above-mentioned results for a completely random sequence of Hénon maps. In addition, for this random system of Hénon maps, we introduce the notion of average Green functions and average Green currents which carry many typical features of the classical Green functions and Green currents. Third part consists of some results about the global dynamics of a special class of skew maps. To prove these results, we use the knowledge of dynamical behavior of pseudo-random sequence of Hénon maps widely. We show that the global skew map is strongly mixing for a class of invariant measures and also provide a lower bound on the topological entropy of the skew product. We conclude the thesis by studying another class of maps which are skew products of holomorphic endomorphisms of \mbb Pk fibered over a compact base. We define the fibered Fatou components and show that they are pseudoconvex and Kobayashi hyperbolic.