Degeneration of quadratic polynomial endomorphisms to a Hénon map

2018/03/28 by Fabrizio Bianchi, Bianchi, Fabrizio, Yûsuke Okuyama +1
Mathematics · Physics and Astronomy · #32H50 #32U40 #37F45 #37H15 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1803.10471

openalex publication_date 2018/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an algebraic family (f_t) of regular quadratic polynomial endomorphisms of ℂ² parametrized by \mathbbD^* and degenerating to a Hénon map at t=0, we study the continuous (and indeed harmonic) extendibility across t=0 of a potential of the bifurcation current on \mathbbD^* with the explicit computation of the non-archimedean Lyapunov exponent associated to (f_t). The individual Lyapunov exponents of f_t are also investigated near t=0. Using (f_t), we also see that any Hénon map is accumulated by the bifurcation locus in the space of quadratic holomorphic endomorphisms of \mathbb P².

Citations

• Discontinuity of a degenerating escape rate
• Lattès maps and the interior of the bifurcation locus

Related

• Lyapunov exponents and bifurcation current for polynomial-like maps
• Hyperbolicity and bifurcations in holomorphic families of polynomial skew products
• Polynomial diffeomorphisms of C²: V. Critical points and Lyapunov exponents
• The Critical Locus for Complex Hénon Maps
• Bifurcation currents in holomorphic dynamics on \bf P^k