20 research outputs found
Post-critical set and non existence of preserved meromorphic two-forms
We present a family of birational transformations in depending on
two, or three, parameters which does not, generically, preserve meromorphic
two-forms. With the introduction of the orbit of the critical set (vanishing
condition of the Jacobian), also called ``post-critical set'', we get some new
structures, some "non-analytic" two-form which reduce to meromorphic two-forms
for particular subvarieties in the parameter space. On these subvarieties, the
iterates of the critical set have a polynomial growth in the \emph{degrees of
the parameters}, while one has an exponential growth out of these subspaces.
The analysis of our birational transformation in is first carried out
using Diller-Favre criterion in order to find the complexity reduction of the
mapping. The integrable cases are found. The identification between the
complexity growth and the topological entropy is, one more time, verified. We
perform plots of the post-critical set, as well as calculations of Lyapunov
exponents for many orbits, confirming that generically no meromorphic two-form
can be preserved for this mapping. These birational transformations in ,
which, generically, do not preserve any meromorphic two-form, are extremely
similar to other birational transformations we previously studied, which do
preserve meromorphic two-forms. We note that these two sets of birational
transformations exhibit totally similar results as far as topological
complexity is concerned, but drastically different results as far as a more
``probabilistic'' approach of dynamical systems is concerned (Lyapunov
exponents). With these examples we see that the existence of a preserved
meromorphic two-form explains most of the (numerical) discrepancy between the
topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure
Green Currents for Meromorphic Maps of Compact K\"ahler Manifolds
We consider the dynamics of meromorphic maps of compact K\"ahler manifolds.
In this work, our goal is to locate the non-nef locus of invariant classes and
provide necessary and sufficient conditions for existence of Green currents in
codimension one.Comment: Statement of Theorem 1.5 is slightly improved. Proposition 5.2 and
Theorem 5.3 are adde
Embeddings of SL(2,Z) into the Cremona group
Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona
group are studied. Infinitely many non-conjugate embeddings which preserve the
type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements
of the same type) are provided. The existence of infinitely many non-conjugate
elliptic, parabolic and hyperbolic embeddings is also shown.
In particular, a group G of automorphisms of a smooth surface S obtained by
blowing-up 10 points of the complex projective plane is given. The group G is
isomorphic to SL(2,Z), preserves an elliptic curve and all its elements of
infinite order are hyperbolic.Comment: to appear in Transformation Group
On the complexity of some birational transformations
Using three different approaches, we analyze the complexity of various
birational maps constructed from simple operations (inversions) on square
matrices of arbitrary size. The first approach consists in the study of the
images of lines, and relies mainly on univariate polynomial algebra, the second
approach is a singularity analysis, and the third method is more numerical,
using integer arithmetics. Each method has its own domain of application, but
they give corroborating results, and lead us to a conjecture on the complexity
of a class of maps constructed from matrix inversions
A birational mapping with a strange attractor: Post critical set and covariant curves
We consider some two-dimensional birational transformations. One of them is a
birational deformation of the H\'enon map. For some of these birational
mappings, the post critical set (i.e. the iterates of the critical set) is
infinite and we show that this gives straightforwardly the algebraic covariant
curves of the transformation when they exist. These covariant curves are used
to build the preserved meromorphic two-form. One may have also an infinite post
critical set yielding a covariant curve which is not algebraic (transcendent).
For two of the birational mappings considered, the post critical set is not
infinite and we claim that there is no algebraic covariant curve and no
preserved meromorphic two-form. For these two mappings with non infinite post
critical sets, attracting sets occur and we show that they pass the usual tests
(Lyapunov exponents and the fractal dimension) for being strange attractors.
The strange attractor of one of these two mappings is unbounded.Comment: 26 pages, 11 figure