Do you want to publish a course? Click here

Dynamical pairs with an absolutely continuous bifurcation measure

88   0   0.0 ( 0 )
 Added by Thomas Gauthier
 Publication date 2018
  fields
and research's language is English




Ask ChatGPT about the research

In this article, we study algebraic dynamical pairs $(f,a)$ parametrized by an irreducible quasi-projective curve $Lambda$ having an absolutely continuous bifurcation measure. We prove that, if $f$ is non-isotrivial and $(f,a)$ is unstable, this is equivalent to the fact that $f$ is a family of Latt`es maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of $(f,a)$ and a similarity property, at any transversely prerepelling parameter $lambda_0$, between the measure $mu_{f,a}$ and the maximal entropy measure of $f_{lambda_0}$. We also establish an equivalent result for dynamical pairs of $mathbb{P}^k$, under an additional assumption.



rate research

Read More

The moduli space $mathcal{M}_d$ of degree $dgeq2$ rational maps can naturally be endowed with a measure $mu_mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $mu_mathrm{bif}$ has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of $mu_mathrm{bif}$ and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.
Let $Lambda$ be a complex manifold and let $(f_lambda)_{lambdain Lambda}$ be a holomorphic family of rational maps of degree $dgeq 2$ of $mathbb{P}^1$. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdins bound of the volume of the image of a dynamical ball. Applying our technics to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of $mathbb{P}^k$.
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andre-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
By generalising Rudins construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.
100 - Yunping Jiang 2010
The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichmuller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichmuller structure. Moreover, this Teichmuller structure is a complete structure and is the completion of the space of smooth potentials under this Teichmuller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا