A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function

摘要

If T-circumflex: X-circumflex→X-circumflex is a two-sided subshift on a finite number of suymbols and T : X → X is the corresponding one-sided subshift we give a necessary and sufficient condition for a continuous function f : X-circumflex → R to be cohomologous, with respect to (X-circumflex, T-circumflex), to a continuous function which can be defined on X. The property of being cohomologous to such a 'one-sided' function is important in the study of equilibrium states because there are extra tools, such as the Ruelle transfer operator, for one-sided systems. We present the results in the more general context when T : X → X is a continuous surjection of a compact metric space and T-circumflex : X-circumflex → X-circumflex is its natural extension, and look at the special case of subshifts. We also prove some related results and give applications to subshifts.