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Journal of Modern Dynamics (JMD)
 

Typical dynamics of plane rational maps with equal degrees

Pages: 353 - 377, Volume 10, 2016      doi:10.3934/jmd.2016.10.353

 
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Jeffrey Diller - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Han Liu - Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States (email)
Roland K. W. Roeder - IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202, United States (email)

Abstract: 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, ergodic properties, dynamical degrees.
Mathematics Subject Classification:  Primary: 37F10; Secondary: 32H50.

Received: January 2016;      Revised: April 2016;      Available Online: August 2016.

 References