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Register a new userA note on two types of Lyapunov exponents and entropies for $mathbb{Z}^k$-actions
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Authors
Yujun Zhu
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In this paper, two types of Lyapunov exponents: random Lyapunov exponents and directional Lyapunov exponents, and the corresponding entropies: random entropy and directional entropy, are considered for smooth $mathbb{Z}^k$-actions. The close relation among these quantities are investigated and the formulas of them are given via the Lyapunov exponents of the generators.
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In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $mathbb{Z}^k$-actions which are generated by random compositions of the generators of $mathbb{Z}^k$-actions. Applying Pesins theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $mathbb{Z}^k$(or $mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and $C^{1}$ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{Z}^k$-actions.
We obtain a sufficient condition for a substitution ${mathbb Z}$-action to have pure singular spectrum in terms of the top Lyapunov exponent of the spectral cocycle introduced in arXiv:1802.04783 by the authors. It is applied to a family of examples, including those associated with self-similar interval exchange transformations.
We study directional mean dimension of $mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+infty]$-valued function on the torus) is not continuous. On the other hand, we prove that if a $mathbb{Z}^k$-action is continuum-wise expansive, then the values of its $(k-1)$-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamotos theorem on mean dimension and expansive multiparameter actions) of a classical result due to Ma~ne: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.
Two different types of perturbations of the Lorenz 63 dynamical system for Rayleigh-Benard convection by multiplicative noise -- called stochastic advection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise -- are found to produce qualitatively different effects, possibly because the total phase-space volume contraction rates are different. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents for the FD noise was found to differ significantly from the value of the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz 63 value with high accuracy. The Lagrangian averaged version of the SALT equations (LA SALT) is found to yield a closed deterministic subsystem for the expected solutions which is found to be isomorphic to the original Lorenz 63 dynamical system. The solutions of the closed chaotic subsystem, in turn, drive a linear stochastic system for the fluctuations of the LA SALT solutions around their expected values.
We study the problem of embedding arbitrary $mathbb{Z}^k$-actions into the shift action on the infinite dimensional cube $left([0,1]^Dright)^{mathbb{Z}^k}$. We prove that if a $mathbb{Z}^k$-action satisfies the marker property (in particular if it is a minimal system without periodic points) and if its mean dimension is smaller than $D/2$ then we can embed it in the shift on $left([0,1]^Dright)^{mathbb{Z}^k}$. The value $D/2$ here is optimal. The proof goes through signal analysis. We develop the theory of encoding $mathbb{Z}^k$-actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of the Euclidean space into a band-limited function which is constructed from another tiling.
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