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Real and complex dynamics of a family of birational maps of the plane: the golden mean subshift. (English) Zbl 1083.37038

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.

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

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