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Register a new userLimit Drift for Complex Feigenbaum Mappings
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We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point
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We study the problem of the existence of wild attractors for critical circle covering maps with Fibonacci dynamics.
Given a discrete subgroup $Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $Bbb{H}^n_{Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $Gamma$ on all of $ Bbb{P}^n_{Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously and can be described as the complement of the union of all complex projective hyperplanes in $ Bbb{P}^n_{Bbb{C}}$ which are tangent to $partial Bbb{H}^n_{Bbb{C}}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(Gamma)$.
We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than~2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.
In this short note, we give a sketch of a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the Feigenbaum fixed point. The proof uses the non existence of invariant line fields in the Feigenbaum tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument, different from previous methods by C. McMullen and M. Lyubich. The method is very general: for instance, it can be used in the classical renormalization operator and in the Fibonacci renormalization operator.
Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincare first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincare map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincare mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rossler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto--Sivashinsky equation.
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