Mathematical Physics X pp 334-340 | Cite as

# A Variational Approach to the Random Diffeomorphisms Type Perturbations of a Hyperbolic Diffeomorphism

Conference paper

## Abstract

Let for the principal eigenvalues

*f*be a diffeomorphism of a compact Riemannian manifold*M*. The standard model of random perturbations of*f*is generated by a particle which jumps from*x*to*fx*and smears with some distribution close to the*δ*— function at*fx*. This model has the continuous time version where a flow is perturbed by a small diffusion. This approach leads to a Markov chain*X*_{ n }^{ ε }(or diffusion*X*_{ t }^{ ε }, in the continuous time case) with a small parameter*ε*> 0 and one is interested whether invariant measures of*X*_{ n }^{ ε }converge as*ε*→ 0 to a particular invariant measure of the diffeomorphism*f*. To describe limiting measures I employed in [5] the Donsker-Varadhan variational formula$$ {\lambda ^\varepsilon }(V) = \mathop {\sup }\limits_{\mu \in p(M)} (\int {Vd\mu } - {I^\varepsilon }(\mu )) $$

(1)

*λ*^{ ε }*(V)*of the operators*P*_{V}^{ε}*g = P*^{ ε }*(e*^{V}*g)*where*P*^{ ε }is the transition operator of the Markov chain*X*_{n}^{ε},*V*is a continuous function, and*I(μ*) is certain lower semicontinuous convex functional on the space*P(M)*of probability measures on*M.*It turns out that if*f*is a hyperbolic diffeomorphism then λ^{ε}(V) converges as*ε*→ 0 to the topological pressure*Q(V + φ*^{ u }) of*f*corresponding to the function*V + φ*^{ u }where φ^{u}= -log*J*^{ u }*(x)*and*J*^{ u }*(x)*is the Jacobian of the differential*Df*restricted to the unstable subbundle.## Keywords

Invariant Measure Random Perturbation Compact Riemannian Manifold Principal Eigenvalue Random Dynamical System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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