The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to haveâ€¦ (More)

We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study isâ€¦ (More)

We study Ruelleâ€“Perronâ€“Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper,â€¦ (More)

We give a brief review of holomorphic motions and its relation with quasiconformal mapping theory. Furthermore, we apply the holomorphic motions to give new proofs of famous KÃ¶nigâ€™s Theorem andâ€¦ (More)

We prove a technical lemma, the C-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a C onedimensional mapping f : M 7â†’ M with finitely many, non-recurrent, power lawâ€¦ (More)

We prove that the period doubling operator has an expanding di rection at the xed point We use the induced operator a Perron Frobenius type operator to study the linearization of the period dou blingâ€¦ (More)

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 HÃ¶lder continuous, whileâ€¦ (More)

We study the geometry of certain one-dimensional maps as dynamical systems. We prove the property of bounded and bounded nearby geometry of certain C one-dimensional maps with finitely many criticalâ€¦ (More)

Following the first part of our research, we prove in this paper that a sub-hyperbolic semi-rational map with infinite post-critical set is combinatorially and locally holomorphically equivalent to aâ€¦ (More)

In this paper, we give a proof of Sullivanâ€™s complex bounds for the Feigenbaum quadratic polynomial and show that the Julia set of the Feigenbaum quadratic polynomial is connected and locallyâ€¦ (More)