Yunping Jiang

Renormalization and geometry in one-dimensional and complex dynamics

Denjoy distortion principle and renormalization Koebe distortion principle geometry of one dimensional mappings renormalization in folding mappings renormalization in quadratic-like maps thermodynamical formalism and renormalization operator.

On Ulam-von Neumann transformations

We define and study Ulam-von Neumann transformations which are certain interval mappings and conjugate toq(x)=1−2x2 on [−1,1]. We use a singular metric on [−1,1] to study a Ulam-von Neumann transformation. This singular metric is universal in the sense that it does not depend on any particular mapping but only on the exponent of this mapping at its unique critical point. We give the smooth classification of Ulam-von Neumann transformations by their eigenvalues at periodic points and exponents and asymmetries.

SMOOTH CLASSIFICATION OF GEOMETRICALLY FINITE ONE-DIMENSIONAL MAPS

The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Holder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete C1-invariants within a mixing topological conjugacy class.

Geometry of geometrically finite one-dimensional maps

We study the geometry of certain one-dimensional maps as dynamical systems. We prove the property of bounded and bounded nearby geometry of certainC1+α one-dimensional maps with finitely many critical points. This property enables us to give the quasisymmetric classification of geometrically finite one-dimensional maps.

Transfer operators acting on Zygmund functions

We obtain a formula for the essential spectral radius Pess Of transfer-type operators associated with families of C1+6 diffeomorphisms of the line and Zygmund, or H6lder, weights acting on Banach spaces of Zygmund (respectively H6lder) functions. In the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive.

Analyticity of the susceptibility function for unimodal Markovian maps of the interval

We study the expression (susceptibility) where f is a unimodal Markovian map of the interval I, ? = ?f is the corresponding absolutely continuous invariant measure and A is a C1 function defined on I. We show that ?(?) is analytic near ? = 1, where ?(1) is formally the derivative of with respect to f in the direction of the vector field X.

α-ASYMPTOTICALLY CONFORMAL FIXED POINTS AND HOLOMORPHIC MOTIONS

The term integrable asymptotically conformal at a point for a quasiconformal map defined on a domain is defined. Furthermore, we prove that there is a normal form for this kind attracting or repelling or super-attracting fixed point with the control condition under a quasicon- formal change of coordinate which is also asymptotically conformal at this fixed point. The change of coordinate is essentially unique. These results generalize Konigs Theorem and Bottchers The- orem in classical complex analysis. The idea in proofs is new and uses holomorphic motion theory and provides a new understanding of the inside mechanism of these two famous theorems too.

Asymptotic Differentiable Structure on Cantor Set

We study hyperbolic maps depending on a parameter ε. Each of them has an invariant Cantor set. As ε tends to zero, the map approaches the boundary of hyperbolicity. We analyze the asymptotics of scaling function of the invariant Cantor set as ε goes to zero. We show that there is a limiting scaling function of the limiting map and this scaling function has dense jump discontinuities because the limiting map is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting map with respect to a Ulam-von Neumann type metric.

Dynamics of certain non-conformal semigroups

A semigroup generated by two dimensional C l+α contracting maps 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 that the shape of the image of the core of a ball under any element of a regular semigroup is good (bounded geometric distortion like the Koebe 1/4-lemma [1]). And we use it to show a lower and an upper bounds of the Hausdorff dimension of the limit set of a regular semigroup. We also consider a semigroup generated by higher dimensional maps.

Markov Partitions and Feigenbaum-like Mappings

We construct a Markov partition for a Feigenbaum-like mapping. We prove that this Markov partition has bounded nearby geometry property similar to that for a geometrically finite one-dimensional mappings [8]. Using this property, we give a simple proof that any two Feigenbaum-like mappings are topologically conjugate and the conjugacy is quasisymmetric.