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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 $mathbb{P}^k$.
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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.
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.
In the present paper, we study the distribution of the return points in the fibers for a RDS (random dynamical systems) nonuniformly expanding preserving an ergodic probability, we also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property between fibers. As consequences, we prove that any expanding measure is the limit of probability measure whose measures of disintegration on the fibers are supported by a finite number of return points and we prove that the average of the measures on the fibers corresponding to a disintegration, along an orbit in the base dynamics is the limit of Dirac measures supported in return orbits on the fibers.
It has been demonstrated that excitable media with a tree structure performed better than other network topologies, it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characterization for entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change.
Let $f:Xto X$ be a dominating meromorphic map of a compact Kahler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $ u$ of large entropy supported by $mathrm{supp}(S)$. Defining dimensions for $ u$ and $S$, we give inequalities `a la Ma~ne involving the Lyapunov exponents of $ u$ and those dimensions. We give dynamical applications of those inequalities.
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