Browsing by Subject "rational maps"

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    COMPUTING DYNAMICAL DEGREES OF RATIONAL MAPS ON MODULI SPACE
    (Cambridge, 2015) Koch, Sarah; Roeder, Roland K.; Department of Mathematical Sciences, School of Science
    The dynamical degrees of a rational map f:X⇢X are fundamental invariants describing the rate of growth of the action of iterates of f on the cohomology of X. When f has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps f:XN⇢XN, where XN is isomorphic to the Deligne–Mumford compactification M¯¯¯¯0,N+3. We exploit the stratified structure of XN to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps f:XN⇢XN, where dim(XN)≤3 and the first dynamical degrees for the mappings where dim(XN)≤5. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.
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    Typical dynamics of plane rational maps with equal degrees
    (AIMS, 2016) Diller, Jeffrey; Liu, Han; Roeder, Roland K. W.; Department of Mathematical Sciences, School of Science
    Let f:CP2⇢CP2 be a rational map with algebraic and topological degrees both equal to d≥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:CP2→CP2, the perturbed map T∘f admits exactly two ergodic measures of maximal entropy logd, one of saddle type and one of repelling type. Neither measure is supported in an algebraic curve, and fT is 'fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation of CP2. In fact, absence of an invariant foliation extends to all T outside a countable union of algebraic subsets of Aut(P2). 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.