Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices
Eric Bedford Kyounghee Kim
Discrete & Continuous Dynamical Systems - A 2008, 21(4): 977-1013 doi: 10.3934/dcds.2008.21.977
We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set $\mathcal{SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to $\mathcal{SC}_q$.
keywords: Birational maps. Degree growth
Pseudo-automorphisms with no invariant foliation
Eric Bedford Serge Cantat Kyounghee Kim
Journal of Modern Dynamics 2014, 8(2): 221-250 doi: 10.3934/jmd.2014.8.221
We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.
keywords: rational 3-folds cohomologically nonhyperbolic dynamical degrees. invariant foliation regular birational map singular foliation Pseudo-automorphism

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