Generalized Entropy Formula

Abstract

The entropy and the Lyapunov exponents provide two different ways of measuring the complexity of the dynamical behavior of a C ² endomorphism f : M ← associated with an invariant measure µ. Generally speaking, the entropy of the system (M, f ,µ) is bounded up by the sum of positive Lyapunov exponents. This is the famous Ruelle inequality introduced in Chapter II. In some cases such as the invariant measure being absolutely continuous with respect to the Lebesgue measure (see Chapter VI), the inequality can become equality. The equality is the notable Pesin’s entropy formula. As we have shown in Chapter VII, Pesin’s entropy formula is equivalent to the SRB property of the invariant measure. Ledrappier and Young [43] presented a generalized entropy formula, which looks like and covers Pesin’s entropy formula, for any Borel probability measure invariant under a C ² diffeomorphism. This result is successfully generalized to random diffeomorphisms [70].

In this chapter we will extend Ledrappier and Young’s result to the case of C ² endomorphisms following the line of [71] (see Theorem IX.1.3).