Pesin entropy formula

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Boris Hasselblatt and Yakov Pesin (2008), Scholarpedia, 3(3):3733. doi:10.4249/scholarpedia.3733 revision #91644 [link to/cite this article]
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1.00 - Boris Hasselblatt

The Pesin entropy formula is a formula according to which the entropy of a measure that is invariant under a dynamical system is given by the total asymptotic expansion rate present in the dynamical system.

Contents

Introduction

The entropy of a Borel probability measure that is invariant under a diffeomorphism \(f\) is a measure of the overall exponential complexity of the orbit structure of \(f\) as "seen" by this measure. This concept originated from an adaptation of a notion in information theory to the theory of dynamical systems. The content of the Pesin Entropy Formula is that in the case of a smooth measure the entropy is given precisely by the total asymptotic exponential rate of expansion under the diffeomorphism, that is, the sum of the positive Lyapunov exponents of the measure. This formula has been refined to give analogous information about invariant measures that are not absolutely continuous when the expansion rates are weighted suitably. This involves connections between entropy and fractal dimension.

The exposition in this entry freely uses notions in hyperbolic dynamics and nonuniform hyperbolicity that can be found in the pertinent entries.

The Margulis-Ruelle Inequality

There are intimate connections between Lyapunov exponents and entropy. The first to be obtained among these an inequality:

Margulis-Ruelle inequality. Let \(M\) be a compact Riemannian manifold and \(f\) a \(C^{1+\alpha}\)-diffeomorphism that preserves a Borel probability measure \(\nu\ .\) Then \[h_\nu(f)\leq\int_M\Sigma(x)d\nu(x),\ .\] Here \(\Sigma\) is the sum of the positive Lyapunov exponents: \[\tag{1} \Sigma(x):=\sum_{\chi^+_i(x)>0}k_i^+(x)\chi_i^+(x)\ .\]


By the variational principle for topological entropy the supremum over invariant Borel probability measures \(\nu\) of the above quantity gives an upper bound for topological entropy. Conversely, this shows that every diffeomorphism of a compact manifold with positive topological entropy has an ergodic invariant Borel probability measure with at least one positive and one negative Lyapunov exponent.

The Pesin Entropy Formula

The Margulis-Ruelle inequality may be strict (such as when \(f\) has a finite hyperbolic nonwandering set), but not for smooth invariant measures:

Pesin Entropy Formula. Let \(M\) be a compact Riemannian manifold and \(f\) a \(C^{1+\alpha}\)-diffeomorphism that preserves a smooth Borel probability measure \(\nu\ .\) Then \[\tag{2} h_\nu(f)=\int_M\Sigma(x)d\nu(x)\ .\]


The content of this formula is that the entropy of a measure is given exactly by the total expansion in the system.

The original proof of the entropy formula uses the full power of nonuniform hyperbolicity techniques (unstable foliations, their absolute continuity, leaf-subordinated partitions, etc.) but there is a somewhat straightforward approach by Mañe which also gives this formula.

The entropy formula was first surmised by Sinai around 1966 in the course of work on ergodic properties of systems closely related to Anosov systems. He communicated it to Margulis who proved the Margulis-Ruelle inequality for the case of volume-preserving systems (this proof was never published but the fact was well-known and accepted in the Moscow dynamics community). While developing the theory of nonuniformly hyperbolic dynamical systems, Pesin realized that this provided the tools for proving the entropy formula, and he published a proof of the hitherto unknown reverse inequality to the Margulis-Ruelle inequality. Sinai reported this result to Ruelle on a visit to Paris, and Ruelle then published the Margulis-Ruelle inequality for arbitrary invariant probability measures, with a view to making the result applicable to infinite-dimensional dynamical systems such as they arise in the study of partial differential equations.

Sinai-Ruelle-Bowen Measures

Remarkably, this formula and the ergodicity results for smooth invariant measures extend to Sinai-Ruelle-Bowen measures, which greatly extends the power of the theory of nonuniformly hyperbolic dynamical systems.

A hyperbolic measure \(\nu\) is said to be a Sinai-Ruelle-Bowen measure for a diffeomorphism \(f\) if it is absolutely continuous on unstable manifolds. This means, roughly, that restricting \(\nu\) to small neighborhoods of a piece of an unstable manifold and keeping this restriction normalized as the neighborhood shrinks down to the small piece of stable manifold one obtains in the limit the normalized Riemannian measure on the piece of unstable manifold. This very assumption makes it possible to extend the arguments from the volume-preserving situation, and one obtains strong ergodic properties:

Ledrappier-Strelcyn Theorem. A Sinai-Ruelle-Bowen measure admits an ergodic decomposition onto countably many pieces of positive measure. These in turn break up into finitely many pieces that are permuted in such a way that the return map is a Bernoulli automorphism, and the Pesin Entropy Formula (2) holds. Moreover, this entropy formula holds only for Sinai-Ruelle-Bowen measures.

As a direct consequence of the definition one also obtains

Theorem. A \(C^{1+\alpha}\) diffeomorphism of a compact smooth manifold has at most countably many Sinai-Ruelle-Bowen measures, and the basin of attraction of every Sinai-Ruelle-Bowen measure has positive Lebesgue measure.

This says that Sinai-Ruelle-Bowen measures are "physical measures" in that there is a positive probability that a point chosen at random (with respect to Lebesgue measure) will display asymptotics that reflect the Sinai-Ruelle-Bowen measure.

Dimension of a Measure

One may try to obtain a geometric understanding of a measure by determining its fractal dimension in a sense. There are several ways of defining such a dimension of a measure (Hausdorff dimension, upper and lower box and information dimensions). In different contexts one or the other of these may be easier to determine or more interesting to find out, and they are all related to the dynamics and particularly to Lyapunov exponents in a deep way. Fortunately, for hyperbolic measures these various notions tend to coincide.

Young Criterion. If \(\nu\) is a Borel probability measure on a compact metric space of finite topological dimension then Hausdorff dimension and upper and lower box and information dimensions all coincide with the pointwise dimension \(d_\nu(x):=\lim_{r\to0}\frac{\nu(B(x,r))}{\log r}\) if the latter exists and is constant \(\nu\)-almost everywhere. (In this case \(\nu\) is said to be exact-dimensional.) Here \(B(x,r)\) is the ball around \(x\) of radius \(r\ .\)

This criterion applies to hyperbolic measures due to the following result:

Theorem. If \(f\) is a \(C^{1+\alpha}\) diffeomorphism of a compact manifold and \(\nu\) is an \(f\)-invariant hyperbolic Borel probability measure with compact support then \(d_\nu=d^s_\nu+d^u_n\) \(\nu\)-almost everywhere. If \(\nu\) is ergodic then it is exact-dimensional (by invariance).

The summands in the last expression are the pointwise dimensions on stable and unstable slices. These are often much easier to compute or estimate than the total dimension, which makes this product decomposition very useful.

The Ledrappier-Young Entropy Formula

For measures \(\nu\) other than Sinai-Ruelle-Bowen measures the entropy still represents the total expansion in the system, but the sum of the Lyapunov exponents has to be tempered to account for expansion that is "missed" by \(\nu\ .\) Instead of multiplying each positive Lyapunov exponent by the marginal dimension of the corresponding subspace \(E^i(x)\ ,\) one now replaces the dimension of \(E^i(x)\) by the pointwise dimension \(d_\nu^i(x)\) of the corresponding subleaf \(W_i(x)\) of the unstable leaf generated by \(E^i(x)\ .\) This amounts to subtracting "wasted expansion" and gives the following:

Ledrappier-Young Entropy formula. For a \(C^2\) diffeomorphism \(f\) of compact manifold and an invariant Borel probability measure \(\nu\) the entropy of \(\nu\) is given by \[h_\nu(f)=\int_M\sum_{\chi_i^+(x)>0}\chi_i^+(x)(d_\nu^i(x)-d_\nu^{i-1}(x))d\nu(x)\ .\]

References

  • L. Barreira and Y. Pesin: Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, 2007
  • Y. Pesin: Dimension theory in dynamical systems. Contemporary views and applications. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997

Internal references

  • Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
  • Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
  • Roy Adler, Tomasz Downarowicz, Michał Misiurewicz (2008) Topological entropy. Scholarpedia, 3(2):2200.


See Also

Dynamical Systems, Entropy, Ergodic Theory, Hyperbolic Dynamics, Invariant Measures, Lyapunov Exponents, Nonuniform Hyperbolicity, Partial Hyperbolicity, Sinai-Ruelle-Bowen Measure

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