Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 951691, 14 pages
http://dx.doi.org/10.1155/2012/951691
Research Article

Differentiability Properties of the Pre-Image Pressure

Kesong Yan,1,2 Fanping Zeng,2,3 and Gengrong Zhang3

1Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
2Department of Mathematics and Computer Science, Liuzhou Teachers College, Liuzhou, Guangxi 545004, China
3Institute of Mathematics, Guangxi University, Nanning, Guangxi 530004, China

Received 8 March 2012; Accepted 26 March 2012

Academic Editor: M. De la Sen

Copyright © 2012 Kesong Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the differentiability properties of the pre-image pressure. For a TDS (X,T) with finite topological pre-image entropy, we prove the pre-image pressure function Ppre(T,•) is Gateaux differentiable at f∈C(X,R) if and only if Ppre(T,•) has a unique tangent functional at f. Also, we obtain some equivalent conditions for Ppre(T,•) to be Fréchet differentiable at f.