Bedford, Eric; Diller, Jeffrey Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift. (English) Zbl 1083.37038 Am. J. Math. 127, No. 3, 595-646 (2005). This extended paper deals with the 1-parameter family of birational mappings of the plane \[ f_a(x,y)=\left(y\frac{x+a}{x-1}, x+a-1\right), \quad a\in\mathbb{R} \] is a parameter. By combining complex intersection theory and techniques from smooth dynamical systems, the authors give an essentially complete account of the behaviour of both wandering and nonwandering orbits. An abstract model for the dynamics of the mapping \(f_a\) on the nonwandering set \(\Omega\) is the so-called golden mean subshift \((\sigma,\Sigma)\). That is, \(\sigma\) is the shift map and \(\Sigma\) is the topological space of bi-infinite sequences of 0’s and 1’s such that “1” is always followed by “0”. The entropy of this subshift is the logarithm of the golden mean \(\Phi=(1+\sqrt 5)/2\). The authors work simultaneously with the real map \(f_a\) and its complexification \(\widetilde f_a\), using the results from the intersection theory of complex subvarieties. Reviewer: Alois Klíč (Praha) Cited in 13 Documents MSC: 37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37B10 Symbolic dynamics 37B40 Topological entropy 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Keywords:complexification; nonwandering set; current; intersection number; closed relation; birational mappings Software:Prune PDFBibTeX XMLCite \textit{E. Bedford} and \textit{J. Diller}, Am. J. Math. 127, No. 3, 595--646 (2005; Zbl 1083.37038) Full Text: DOI arXiv