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Register a new userOn the genericity of positive exponents of conservative skew products with two-dimensional fibers
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In this paper we study the existence of positive Lyapunov exponents for three different types of skew products, whose fibers are compact Riemannian surfaces and the action on the fibers are by volume preserving diffeomorphisms. These three types include skew products with a volume preserving Anosov diffeomorphism on the basis; or with a subshift of finite type on the basis preserving a measure with product structure; or locally constant skew products with Bernoulli shifts on the basis. We prove the $C^1$-density and $C^r$-openess of the existence of positive Lyapunov exponents on a set of positive measure in the space of such skew products.
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We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior.As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.
We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $Phi(z_1,z_2) = (phi(z_1,z_2), z_2)$. If $phi$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of Mobius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $Phi$ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $mathbb{P}^2(mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation current. This is established in the general setting of families of endomorphisms of $mathbb{P}^k$ and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one.
In this paper we give an elementary treatment of the dynamics of skew tent maps. We divide the two-parameter space into six regions. Two of these regions are further subdivided into infinitely many regions. All of the regions are given explicitly. We find the attractor in each subregion, determine whether the attractor is a periodic orbit or is chaotic, and also determine the asymptotic fate of every point. We find that when the attractor is chaotic, it is either a single interval or the disjoint union of a finite number of intervals; when it is a periodic orbit, all periods are possible. Sometimes, besides the attractor, there exists an invariant chaotic Cantor set.
We prove that in an open and dense set, Symplectic linear cocycles over time one maps of Anosov flows, have positive Lyapunov exponents for SRB measures.
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