We return to the thermodynamic formalism constructions for random expanding in average transforma... more We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron–Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.
We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}... more We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}F\left ( \xi \left ( n \right ),\xi \left ( 2n \right ),...,\xi \left ( \ell n \right ) \right )$ where $\xi \left ( n \right )$ is a stochastic process or a dynamical system. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple recurrence properties of corresponding stochastic processes and dynamical systems. Among our results are: central limit theorem, a.s. central limit theorem, local limit theorem, large deviations and averaging. Some multifractal type questions and open problems will be discussed, as well. Keywords : limit theorems - nonconventional sums - multiple recurrence
In this chapter we shall study the asymptotical behavior of invariant measures, entropies, and ot... more In this chapter we shall study the asymptotical behavior of invariant measures, entropies, and other characteristics of random perturbations of dynamical systems with complicated dynamics satisfying certain hyperbolicity or expanding conditions.
Let f be a diffeomorphism of a compact Riemannian manifold M. The standard model of random pertur... more Let 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 e (or diffusion X t e , in the continuous time case) with a small parameter e > 0 and one is interested whether invariant measures of X n e converge as e → 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) for the principal eigenvalues λ e (V) of the operators P V e g = P e (e V g) where P e is the transition operator of the Markov chain X n e , 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 λe(V) converges as e → 0 to the topological pressure Q(V + φ u ) of f corresponding to the function V + φ u where φu = -log J u (x) and J u (x) is the Jacobian of the differential Df restricted to the unstable subbundle.
We return to the thermodynamic formalism constructions for random expanding in average transforma... more We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron–Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.
We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}... more We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}F\left ( \xi \left ( n \right ),\xi \left ( 2n \right ),...,\xi \left ( \ell n \right ) \right )$ where $\xi \left ( n \right )$ is a stochastic process or a dynamical system. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple recurrence properties of corresponding stochastic processes and dynamical systems. Among our results are: central limit theorem, a.s. central limit theorem, local limit theorem, large deviations and averaging. Some multifractal type questions and open problems will be discussed, as well. Keywords : limit theorems - nonconventional sums - multiple recurrence
In this chapter we shall study the asymptotical behavior of invariant measures, entropies, and ot... more In this chapter we shall study the asymptotical behavior of invariant measures, entropies, and other characteristics of random perturbations of dynamical systems with complicated dynamics satisfying certain hyperbolicity or expanding conditions.
Let f be a diffeomorphism of a compact Riemannian manifold M. The standard model of random pertur... more Let 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 e (or diffusion X t e , in the continuous time case) with a small parameter e > 0 and one is interested whether invariant measures of X n e converge as e → 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) for the principal eigenvalues λ e (V) of the operators P V e g = P e (e V g) where P e is the transition operator of the Markov chain X n e , 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 λe(V) converges as e → 0 to the topological pressure Q(V + φ u ) of f corresponding to the function V + φ u where φu = -log J u (x) and J u (x) is the Jacobian of the differential Df restricted to the unstable subbundle.
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Papers by Yuri Kifer