Abstract
By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $C^{2}$ $\mathbb{N}^2$-actions.
Highlights
The significance of Pesin’s entropy formula lies in its characterizing SRB measures by their Lyapunov exponents and entropy [10]
Entropy formula and SRB measures for random transformations generated by finitely commutative transformations in infinite dimensional Hilbert spaces via its generators, which can be viewed as a generalization of the work in [7, 8, 24, 25] to the infinite dimension spaces
For more recent progress of SRB measures in infinite dimensional spaces, we refer to the elegant survey [23]
Summary
The significance of Pesin’s entropy formula (or Ledrappier-Young’s entropy formula for SRB measures) lies in its characterizing SRB measures by their Lyapunov exponents and entropy [10]. Pesin’s entropy formula for random transformations and stochastic flows of diffeomorphisms in finite dimensional compact spaces were established in [2, 11, 16, 9]. To obtain the relations of metric entropy of the random transformation and the Lyapunov exponents of its generators, the basic strategy is to estimate the random exponential expanding rate in a deterministic subspace by exponential expanding rates of generators in this subspace. By comparing the dynamics of the random transformation with the dynamics of its generators, we reformulate Ruelle’s entropy inequality (Theorem B), the Pesin’s entropy formula and SRB measures (Theorem C) via the generators. Let X be a separable infinite dimensional Hilbert space with inner product < ·, · >, norm · , distance function d and σ-algebra B of Borel sets
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