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Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states

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Abstract

Suppose thatL is a second order elliptic differential operator on a manifoldM, B is a vector field, andV is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior asɛ → 0 of the principal eigenvalueλ ε(V) for the operatorL ε = ɛL + (B, ∇) +V considered on a compact manifold or in a bounded domain with zero boundary conditions. Under certain hyperbolicity conditions on invariant sets of the dynamical system generated by the vector fieldB the limit asɛ → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition asɛ → 0 from Donsker-Varadhan’s variational formula for principal eigenvalues to the variational principle for the topological pressure and unifies previously separate results on random perturbations of dynamical systems.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Yuri Kifer

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  1. Yuri Kifer
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This work was supported by US-Israel Binational Science Foundation.

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Kifer, Y. Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Israel J. Math. 70, 1–47 (1990). https://doi.org/10.1007/BF02807217

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  • DOI: https://doi.org/10.1007/BF02807217

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