De-Jun Feng

Some dimensional results for homogeneous Moran sets

Let ℳ(¦nk¦k⩾1,¦ck¦k⩾1) be the collection of homogeneous Moran sets determined by ¦nk¦k⩾1 and ¦ck¦k⩾1, where ¦nk¦k⩾1 is a sequence of positive integers and ¦ck¦k⩾1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in ℳ are determined. The result is proved that for any values between the maximal and minimal values, there exists an element in ℳ(¦nk¦k⩾1,¦ck¦k⩾1) such that its Hausdorff dimension is equal tos. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map is not upper-semi continuous.

On the Distribution of Long-Term Time Averages on Symbolic Space

The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function Φ to its long-term time averages through the Hausdorff and packing dimensions of the subsets on which Φ has prescribed long-term time-average values. Functions Φ with values in ℝd are considered. For those Φ depending only on finitely many symbols, we get complete results, unifying and completing many partial results.

The variational principle for products of non-negative matrices

Let (ΣA, σ) be a subshift of finite type and let M(x) be a continuous function on ΣA taking values in the set of non-negative matrices. We set up the variational principle between the pressure function, entropy and Lyapunov exponent for M on ΣA. We also present some properties of equilibrium states.

The Hausdorff dimension of recurrent sets in symbolic spaces

Let (�, σ ) be the one-sided shift space on m symbols. For any x = (xi)i1 ∈ � and positive integer n, define Rn(x) = inf{j n : x1x2 ··· xn = xj +1xj +2 ··· xj +n}. We prove that for each pair of numbers α, β ∈ [0, ∞] with α β, the following recurrent set Eα,β = � x ∈ � : lim inf n→∞ log Rn(x) n = α, lim sup n→∞ log Rn(x) n = β

Gibbs properties of self-conformal measures and the multifractal formalism

We prove that for any self-conformal measures, without any separation conditions, the multifractal formalism partially holds. The result follows by establishing certain Gibbs properties for self-conformal measures.

Equilibrium states of the pressure function for products of matrices

Let

Generation of finite tight frames by Householder transformations

\{M_i\}_{i=1}^\ell

A property of Pisot numbers

be a non-trivial family of

Dimensions of random covering sets in Riemann manifolds

d\times d