arXiv subjects

Yunping Jiang

Publications and source records attributed to Yunping Jiang.

43 records · Page 3Linked to original sources

arXiv1991

On the quasisymmetrical classification of infinitely renormalizable maps: I. Maps with Feigenbaum's topology.

A semigroup (dynamical system) generated by $C^{1+α}$-contracting mappings is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives of generators and the smoothness $α$ of the generators satisfy a compatibility condition $K< 1/l^α$. We prove the {\em geometric distortion lemma} for a regular semigroup generated by $C^{1+α}$-contracting mappings.

math.DS↗

arXiv1991

On the quasisymmetrical classification of infinitely renormalizable maps: II. remarks on maps with a bounded type topology.

We use the upper and lower potential functions and Bowen's formula estimating the Hausdorff dimension of the limit set of a regular semigroup generated by finitely many $C^{1+α}$-contracting mappings. This result is an application of the geometric distortion lemma in the first paper at this series.

math.DS↗

arXiv1991

Dynamics of certain smooth one-dimensional mappings III: Scaling function geometry

We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the critical points of a geometrically finite one-dimensional mapping form a complete set of $C^{1}$-invariants within a topological conjugacy class.

math.DS↗

arXiv1991

Dynamics of certain smooth one-dimensional mappings IV: Asymptotic geometry of Cantor sets

We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of them has an invariant Cantor set. As $\varepsilon $ tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function geometry of the invariant Cantor set as $\varepsilon $ goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed $\sqrt {\varepsilon}$. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting mapping with respect to the Ulam-von Neumann type metric.

math.DS↗

arXiv1991

Expanding direction of the period doubling operator

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a ``Perron-Frobenius type operator'', to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.

math.DS↗

arXiv1991

Dynamics of certain smooth one-dimensional mappings I: The $C^{1+α}$-Denjoy-Koebe distortion lemma

We prove a technical lemma, the $C^{1+α}$-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a $C^{1+α}$ one-dimensional mapping $f:M\mapsto M$ with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.

math.DS↗

arXiv1991

Dynamics of certain smooth one-dimensional mappings II: geometrically finite one-dimensional mappings

We study geometrically finite one-dimensional mappings. These are a subspace of $C^{1+α}$ one-dimensional mappings with finitely many, critically finite critical points. We study some geometric properties of a mapping in this subspace. We prove that this subspace is closed under quasisymmetrical conjugacy. We also prove that if two mappings in this subspace are topologically conjugate, they are then quasisymmetrically conjugate. We show some examples of geometrically finite one-dimensional mappings.

math.DS↗