A New Definition of t-Entropy for Transfer Operators

A new definition of t-entropy for transfer operators The article presents a new definition of t-entropy that makes it more explicit and simplifies the process of its calculation. In the series of articles [1, 2, 3, 4, 5, 6] there have been established the variational principles for the spectral radii of weighted shift and transfer operators generated by an arbitrary dynamical system. These principles are based on the Legendre duality and the main role here is played by a newly introduced dynamical invariant — t-entropy, which gives the explicit form of the Legendre dual object to the logarithm of the spectral radii of operators in question. The description of t-entropy is not elementary and its calculation is rather sophisticated. In the present article we give a new definition of t-entropy that makes it more explicit and essentially simplifies the process of its calculation. The article consists of two sections. In Section 1 we consider t-entropy for the model example of transfer operators associated with continuous dynamical systems. The new definition of t-entropy is introduced here in Theorem 2. In Section 2 we discuss the general C *-dynamical situation. To illustrate similarity and difference between the objects considered in the model and general situations we present here a number of examples and finally introduce the general new definition of t-entropy in Theorem 10. 1 A new definition of t-entropy for continuous dynamical systems Let us consider a Hausdorff compact space X. We denote by C(X) the algebra of continuous real-valued functions on X equipped with the uniform norm. Let α : X → X be a continuous mapping. This mapping generates the dynamical system with discrete time, which will be denoted by (X, α).