Let T be a C^2-expanding self-map of a compact, connected, smooth, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x in M, the set of points whose forward orbit closures miss x is a winning set.
We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an $L^2$ kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subje…
We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions).
A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.
We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion we construct new Markov rectangles such that their crossections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain Hölder functions. The results are based on the methods developed by Sarig.
We give an explicit geometric description of the $\times2,\times3$ system, and use his to study a uniform family of Markov partitions related to those of Wilson and Abramov. The behaviour of these partitions is stable across expansive cones and transitions in this behaviour detects the non-expansive lines.
Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby hyperbolic components in terms of the symbolic dynamics. This means we can count them, and also obtain topological information. We also determine extra conditions under which all nearby type IV hyperbolic components are given by matings. These are probably the first known results of this type.
This monographic short book is intended to a brief introduction in the classic topic of the theory of Hiperbolic Dynamical Systems, for Spanish speaking students of an undergraduate course in Mathematics. We revisit the classic definition of Markov Partitions for Anosov diffeomorphisms on compact manifolds. We focus on the theorem of Sinai (Func. Anal. and its Applic. 2, 1968) stsating the existence of such partitions with arbitrarily small diameter. To prove the theorem of Sinai we restate the constructive method in Bowen, Lecture Notes in Math. 470, 1975. Finally we also relate the results, …
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…
In the paper we give an introduction to Anosov diffeomorphisms, ways to represent their chaotic properties and some historical remarks on this subject. A complete classification of hyperbolic linear automorphisms of 2-torus is presented. We introduce a notion of pre-Markov partition for such automorphisms and give their classification and an algorithm for their construction.
For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentia…
Markov partitions designed by Sinai(1968) and Bowen(1970) proved to be an efficient tool for descibing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular, for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai(1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich-Chernov-Sinai, 1990), Markov towers (Young, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai's original idea is surveyed i…
In the present paper we investigate two special types of non-normal numbers. On the one hand we call a number extremely non-normal if the set of accumulation points of its frequency of blocks vector is the full set of shift invariant probability vectors. On the other hand we call a number having maximal oscillation frequency if for any fixed block the set of accumulation points of its frequency vector is the full set of possible probability vectors. The goal is to investigate the Baire category of these numbers for infinite Markov partitions.
It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are very abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a rigorous construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.
We build a Markov partition of the Feigenbaum dynamical plane, and a system of "external rays", which are invariant under rescaling. Applications to geometry of the Feigenbaum map are given.
We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n).
The well-known Hammersley-Clifford theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley-Clifford theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce "Markov cocycles", reparametrisations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley-Clifford theorem for a family of Markov fields which ar…
We define a partition $\mathcal{P}_0$ and a $\mathbb{Z}^2$-rotation ($\mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition $\mathcal{P}_\mathcal{U}$ and a $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $\mathcal{P}_\mathcal{U}$ is a Markov partition for the $\mathbb{Z}^2$-rotation on $\math…
The aim of this survey is twofold. First we show how the Markov tower construction is applicable for obtaining finer stochastic properties, like a local limit theorem of probability theory. Here the fundamental method is the study of the spectrum of the Fourier transform of the Perron--Frobenius operator. These ideas and results are applicable to all systems Young has been considering. Second, we survey the problem of recurrence of the planar Lorentz process. As an application of the results from the first part, we obtain a dynamical proof of recurrence for the finite horizon case. Here basica…
We prove that transport in the phase space of the "most strongly chaotic" dynamical systems has three different stages. Consider a finite Markov partition (coarse graining) $ξ$ of the phase space of such a system. In the first short times interval there is a hierarchy with respect to the values of the first passage probabilities for the elements of $ξ$ and therefore finite time predictions can be made about which element of the Markov partition trajectories will be most likely to hit first at a given moment. In the third long times interval, which goes to infinity, there is an opposite hierarc…
We consider chaotic (hyperbolic) dynamical systems which have a generating Markov partition. Then, open dynamical systems are built by making one element of a Markov partition a hole through which orbits escape. We compare various estimates of the escape rate which correspond to a physical picture of leaking in the entire phase space. Moreover, we uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Exact computations are present for the skewed tent map and Arnold's cat map.
Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient…
