American Institute of Mathematical Sciences

Journals

DCDS
Discrete & Continuous Dynamical Systems - A 2005, 13(5): 1187-1201 doi: 10.3934/dcds.2005.13.1187
In the study of systems which combine slow and fast motions which depend on each other (fully coupled setup) whenever the averaging principle can be justified this usually can be done only in the sense of $L^1$-convergence on the space of initial conditions. When fast motions are hyperbolic (Axiom A) flows or diffeomorphisms (as well as expanding endomorphisms) for each freezed slow variable this form of the averaging principle was derived in [19] and [20] relying on some large deviations arguments which can be applied only in the Axiom A or uniformly expanding case. Here we give another proof which seems to work in a more general framework, in particular, when fast motions are some partially hyperbolic or some nonuniformly hyperbolic dynamical systems or nonuniformly expanding endomorphisms.
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DCDS
Discrete & Continuous Dynamical Systems - A 2018, 38(6): 2687-2716 doi: 10.3934/dcds.2018113

The paper is primarily concerned with the asymptotic behavior as $N\to∞$ of averages of nonconventional arrays having the form ${N^{ - 1}}\sum\limits_{n = 1}^N {\prod\limits_{j = 1}^\ell {{T^{{P_j}(n,N)}}} } {f_j}$ where $f_j$'s are bounded measurable functions, $T$ is an invertible measure preserving transformation and $P_j$'s are polynomials of $n$ and $N$ taking on integer values on integers. It turns out that when $T$ is weakly mixing and $P_j(n, N) = p_jn+q_jN$ are linear or, more generally, have the form $P_j(n, N) = P_j(n)+Q_j(N)$ for some integer valued polynomials $P_j$ and $Q_j$ then the above averages converge in $L^2$ but for general polynomials $P_j$ of both $n$ and $N$ the $L^2$ convergence can be ensured even in the "conventional" case $\ell = 1$ only when $T$ is strongly mixing while for $\ell>1$ strong $2\ell$-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers $\Lambda$ with positive upper density there exists a subset ${\cal N}_\Lambda$ of positive integers having uniformly bounded gaps such that for $N∈{\cal N}_\Lambda$ and at least $\varepsilon N, \, \varepsilon >0$ of $n$'s all numbers $p_jn+q_jN, \, j = 1, ..., \ell,$ belong to $\Lambda$. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.

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DCDS
Discrete & Continuous Dynamical Systems - A 1997, 3(4): 457-476 doi: 10.3934/dcds.1997.3.457
I consider discretized random perturbations of hyperbolic dynamical systems and prove that when perturbation parameter tends to zero invariant measures of corresponding Markov chains converge to the Sinai-Bowen-Ruelle measure of the dynamical system. This provides a robust method for computations of such measures and for visualizations of some hyperbolic attractors by modeling randomly perturbed dynamical systems on a computer. Similar results are true for discretized random perturbations of maps of the interval satisfying the Misiurewicz condition considered in [KK].
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DCDS
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 131-164 doi: 10.3934/dcds.2008.22.131
We introduce a relative Gurevich pressure for random countable topologically mixing Markov shifts. It is shown that the relative variational principle holds for this notion of pressure. We also prove a relative Ruelle-Perron-Frobenius theorem which enables us to construct a wealth of invariant Gibbs measures for locally fiber Hölder continuous functions. This is accomplished via a new construction of an equivariant family of fiber measures using Crauel's relative Prohorov theorem. Some properties of the Gibbs measures are discussed as well.
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DCDS
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 593-594 doi: 10.3934/dcds.2015.35.593
We correct a flaw in the proof of Proposition 6.3 in [1].
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