Minoru Ogawa
We present a sufficient condition that a pseudointegrable system has weak mixing property. The result is derived from Veech's weak mixing theorem for interval exhange [Veech, W.A. Amer.J.Math. {\bf 106}, 1331 (1984)]. We also present an example whose weak mixing property can be proved by the result.
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Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to Hölder-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved in 2011 by D.-J. Feng and A. Käenmäki. In a recent article the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli m…
In ergodic theory, given sufficient conditions on the system, every weak mixing $\mathbb{N}$-action is strong mixing along a density one subset of $\mathbb{N}$. We ask if a similar statement holds in topological dynamics with density one replaced with thickness. We show that given sufficient initial conditions, a group action in topological dynamics is strong mixing on a thick subset of the group if and only if the system is $k$-transitive for all $k$, and conclude that an analogue of this statement from ergodic theory holds in topological dynamics when dealing with abelian groups.
We prove that if $g\geq 2$ then the set of all Abelian differentials $(M,ω)$ for which the vertical flow is mildly mixing is dense in every stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient condition for special flows over irrational rotations and under piecewise constant roof functions to be mildly mixing.
Several stronger forms of transitivity in non-autonomous discrete dynamical systems
本文在非自治离散动力系统中引入并研究了多种更强的传递性概念，证明了其迭代不变性，给出了充分必要条件，并通过反例揭示了与自治系统的差异，最后建立了等价性条件。
It is an open question in smooth ergodic theory whether there exists a Hamiltonian disk map with zero topological entropy and (strong) mixing dynamics. Weak mixing has been known since Anosov and Katok first constructed examples in 1970. Currently all known examples with weak mixing are irrational pseudo-rotations with Liouvillean rotation number on the boundary. Our main result however implies that for a dense subset of Liouville numbers (strong) mixing cannot occur. Our approach involves approximating the flow of a suspension of the given disk map by pseudoholomorphic curves. Ellipticity of …
A Krieger Embedding Theorem for Near Markov Sofic Shifts
本文推广了Krieger嵌入定理，给出了将子平移嵌入到混合（不可约）近马尔可夫索菲克平移的充要条件，并证明了当被嵌入子平移为不可约索菲克时，条件可有限判定。
Sufficient conditions for distributional chaos of type I
本文研究了I型分布混沌（DC1），将混合Anosov流的时间一映射表现出DC1的事实进行推广，得出了DC1的简单充分条件。
We give sufficient Gordin-type criteria for the iterated (enhanced) weak invariance principle to hold for deterministic dynamical systems. Such an invariance principle is intrinsically related to the interpretation of stochastic integrals. We illustrate this with examples of deterministic fast-slow systems where our iterated invariance principle yields convergence to a stochastic differential equation.
We consider special flows over the rotation by an irrational $α$ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and the sum of jumps $\neq 0$. We show that all such flows are weakly mixing. Under the additional assumption that $α$ has bounded partial quotients, we study weak Ratner's property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is classical that the flows under consideration are not mixing, one more condition on the s…
We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integrals with random integrands.
Quantitative exponential mixing for the randomized Chirikov standard map
本文研究了随机Chirikov标准映射在二维环面上的混合性质，通过建立不可压缩随机动力系统准则，证明了在激励强度足够大时，随机映射具有几乎必然的定量指数混合性，并给出了更温和的参数条件以获得定性指数混合性和增强的耗散性。
Verification of mixing properties in two-dimensional shifts of finite type
针对二维有限型移位难以系统判定混合程度的问题，提出基于条带转移矩阵、连接算子及可扩展条件的有限步验证方法，可证明拓扑混合与强 specification。
We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that partition-regular system of linear equations with integer coefficients is solvable in any WM set.
We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok-Stepin, Veech, and Avila-Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have…
The results of this paper are 3-folded. Firstly, for any stationary determinantal process on the integer lattice, induced by strictly positive and strictly contractive involution kernel, we obtain the necessary and sufficient condition for the $ψ$-mixing property. Secondly, we obtain the existence of the $L^q$-dimensions of the stationary determinantal measure on symbolic space $\{0, 1\}^\mathbb{N}$ under appropriate conditions. Thirdly, the previous two results together imply the precise increasing rate of the longest common substring of a typical pair of points in $\{0, 1\}^\mathbb{N}$.
Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show that \[ \{0<t<1\mid f_{t} \text{is weakly mixing}\} \] is a residual subset of $X$ of full Lebesgue measure. The result is proved by establishing a generic Diophantine sufficient condition on $t$ for $f_{t}$ to be weakly mixing.
In the context of stationary $\mathbb{Z}^d$ nearest-neighbour Gibbs measures $μ$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property (TSSM)) on the support of $μ$ sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of TSSM for studying strong spatial mixing on systems with hard constraints. We also show that TSSM is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by D. Gamarnik and D…
For a strictly stationary sequence of random vectors in $\mathbb{R}^d$ we study convergence of partial sum processes to Lévy stable process in the Skorohod space with $J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.
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