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 that is not upper-semi continuous.Comment: 44 page

Subdifferential and optimality conditions for the difference of set-valued mappings

In this paper, an existence theorem of the subgradients for set-valued mappings, which introduced by Borwein (Math Scand 48:189-204, 1981), and relations between this subdifferential and the subdifferential introduced by Baier and Jahn (J Optim Theory Appl 100:233-240, 1999), are obtained. By using the concept of this subdifferential, the sufficient optimality conditions for generalized D. C. multiobjective optimization problems are established. And the necessary optimality conditions, which are the generalizations of that in Gadhi (Positivity 9:687-703, 2005), are also established. Moreover, by using a special scalarization function, a real set-valued optimization problem is introduced and the equivalent relations between the solutions are proved for the real set-valued optimization problem and a generalized D. C. multiobjective optimization problem