Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence
This Volume
On small gaps in the length spectrum
2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353
Typical dynamics of plane rational maps with equal degrees
Jeffrey Diller 1, , Han Liu 1, and Roland K. W. Roeder 2,
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States
IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202, United States
Received January 2016 Revised April 2016 Published August 2016
Let $f:\mathbb{CP}^2⇢\mathbb{CP}^2$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle type and one of repelling type. Neither measure is supported in an algebraic curve, and $f_T$ is 'fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation of $\mathbb{C}\mathbb{P}^2$. In fact, absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets of $Aut(\mathbb{P}^2)$. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.
Keywords: Rational maps, dynamical degrees., ergodic properties.
Mathematics Subject Classification: Primary: 37F10; Secondary: 32H5.
Citation: Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353
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Jeffrey Diller Han Liu Roland K. W. Roeder