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

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