A new entropy formula of Ledrappier–Young type for linear toral dynamics

A detailed Ledrappier–Young theory is presented for linear toral dynamics. First, a proof is given for a simplified definition for local entropies which holds in more general settings besides the current linear dynamics. Then it is shown that the transverse dimensions can be defined directly via the Smale structure of the linear dynamical system. A new entropy formula of Ledrappier–Young type is obtained. The conjecture of Ledrappier and Xie [Vanishing transverse entropy in smooth ergodic theory. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1229–1235] is also discussed for such linear dynamics.

Vanishing transverse entropy in smooth ergodic theory

For a measure-preserving transformation, the entropy being zero means that there is no increasingσ-algebra. In this note, we prove that a similar phenomenon occurs forC2diffeomorphisms when considering the increment between the partial entropies associated with different exponents.