arXiv subjects

Denis Gaidashev

Publications and source records attributed to Denis Gaidashev.

23 records · Page 2Linked to original sources

arXiv2010

On Analytic Perturbations of a Family of Feigenbaum-like Equations

We prove existence of solutions $(ϕ,λ)$ of a family of of Feigenbaum-like equations \label{family} ϕ(x)={1+\eps \over λ} ϕ(ϕ(λx)) -\eps x +τ(x), where $\eps$ is a small real number and $τ$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\fR$. The family $(\ref{family})$ appears in the context of period-doubling renormalization for area-preserving maps (cf. \cite{GK}). Our proof is a development of ideas of H. Epstein (cf \cite{Eps1}, \cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms $\eps x +τ(x)$ in the equation $(\ref{family})$. The method relies on a construction of novel {\it a-priori} bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter $λ$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function.

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arXiv2010

Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a "universal" area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider {\it infinitely renormalizable} maps -- maps on the renormalization stable manifold in some neighborhood of $F_*$ -- and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_*$ we prove the existence of a "stable" invariant set $\cC^\infty_F$ such that the maximal Lyapunov exponent of $F \arrowvert_{\cC^\infty_F}$ is zero, and whose Hausdorff dimension satisfies $${\rm dim}_H(\cC_F^{\infty}) \le 0.5324.$$ We also show that there exists a submanifold, $\bW_ω$, of finite codimension in the renormalization local stable manifold, such that for all $F\in\bW_ω$ the set $\cC^\infty_F$ is "weakly rigid": the dynamics of any two maps in this submanifold, restricted to the stable set $\cC^\infty_F$, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.

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arXiv2009

Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a "universal" area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider maps in some neighbourhood of $F_*$ and study their dynamics. We first demonstrate that the map $F_*$ admits a "bi-infinite heteroclinic tangle": a sequence of periodic points $\{z_k\}$, $k \in \fZ$, |z_k| \converge{k \to \infty} 0, \quad |z_k| \converge{k \to -\infty} \infty, whose stable and unstable manifolds intersect transversally; and, for any $N \in \fN$, a compact invariant set on which $F_*$ is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of $N$ symbols. A corollary of these results is the existence of {\it unbounded} and {\it oscillating} orbits. We also show that the third iterate for all maps close to $F_*$ admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: $$ 0.7673 \ge {\rm dim}_H(\cC_F) \ge \varepsilon \approx 0.00044 e^{-1797}.$$

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arXiv2009

Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem

It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: $$H^*(x,u)=(ϕ(x)-u,x-ϕ(ϕ(x)-u)),$$ where $ϕ$ solves $$ϕ(x)={2 \over λ} ϕ(ϕ(λx))-x.$$ We then give a ``proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution. The ``proof'' consists of an analytic argument for factorized inverse branches of $ϕ$ together with verification of several inequalities and inclusions of subsets of $\field{C}$ numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.

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arXiv2006

Cylinder renormalization of Siegel disks

We study one of the central open questions in one-dimensional renormalization theory -- the conjectural universality of golden-mean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second author. Numerical implementation of this approach relies on the Constructive Measurable Riemann Mapping Theorem proved by the first author. Our numerical study yields a convincing evidence to support the Hyperbolicity Conjecture in this setting.

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