Periodicities in linear fractional recurrences: degree growth of birational surface maps. (English) Zbl 1171.37023
The authors study periodicity of linear fracitonal recurrences and degree growth of birational surface mappings. They start with the family of birational transformations of the plane defined in affine coordinates by
\[
f(x,y)= \Biggl(y,{a_0+ a_1 x+ a_2 y\over b_0+ b_1x+ b_2 y}\Biggr),\quad a= (a_0,a_1,a_2),\;b= (b_0, b_1,b_2),
\]
or its canonically induced map \(f_{a,b}: \mathbb{P}^2\to \mathbb{P}^2\). The authors study the degree growth of the iterates \(f_{a,b}\). For a generic choice of parameters \(a\) and \(b\), \(f_{a,b}\) is not birationally conjugate to an automorphism. The remaining possibilities are thoroughly classified.
This family of mappings are usually used to construct examples with interesting dynamics.
This family of mappings are usually used to construct examples with interesting dynamics.
Reviewer: Viorel Vâjâitu (Bucureşti)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
| 37B40 | Topological entropy |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
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