The canonical quantization of any hyperbolic symplectomorphism A of the 2-torus yields a periodic unitary operator on a JV-dimensional Hubert space, N =•£. We prove that this quantum system becomes ergodic and mixing at the classical limit (N -• oo, N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly spread in phase space. Contents 1. Von Neumann definition of the quantum ergodicity and mixing properties. Statement of the main results 473 2. Koopman operator on invariant lattices and periodic orbits. Limits of atomic invariant measures supported on periodic orbits via Kloosterman sums 477 3. Quantization of toral automorphisms. Discrete Wigner functions. Support on classical periodic orbits, relation with the Koopman operator and explicit construction of the quantum eigenvectors 482 4. Classical limit of the matrix elements of the observables via Weil-Deligne exponential sums. Weak-* convergence of the Wigner functions. Proof of the main results 495 Appendix A. Some basic results out of number theory 503

The complexity of human interactions with social and natural phenomena is mirrored in the way we describe our experiences through natural language. In order to retain and convey such a high dimensional information, the statistical properties of our linguistic output has to be highly correlated in time. An example are the robust observations, still largely not understood, of correlations on arbitrary long scales in literary texts. In this paper we explain how long-range correlations flow from highly structured linguistic levels down to the building blocks of a text (words, letters, etc..). By combining calculations and data analysis we show that correlations take form of a bursty sequence of events once we approach the semantically relevant topics of the text. The mechanisms we identify are fairly general and can be equally applied to other hierarchical settings.complex systems | language dynamics | long correlations | statistical physics | burstiness L iterary texts are an expression of the natural language ability to project complex and high-dimensional phenomena into a one-dimensional, semantically meaningful sequence of symbols. For this projection to be successful, such sequences have to encode the information in form of structured patterns, such as correlations on arbitrarily long scales (1, 2). Understanding how language processes long-range correlations, an ubiquitous signature of complexity present in human activities (3-7) and in the natural world (8-11), is an important task towards comprehending how natural language works and evolves. This understanding is also crucial to improve the increasingly important applications of information theory and statistical natural language processing, which are mostly based on short-range-correlations methods (12-15).Take your favorite novel and consider the binary sequence obtained by mapping each vowel into a 1 and all other symbols into a 0. One can easily detect structures on neighboring bits, and we certainly expect some repetition patterns on the size of words. But one should certainly be surprised and intrigued when discovering that there are structures (or memory) after several pages or even on arbitrary large scales of this binary sequence. In the last twenty years, similar observations of long-range correlations in texts have been related to large scales characteristics of the novels such as the story being told, the style of the book, the author, and the language (1, 2, 16-21). However, the mechanisms explaining these connections are still missing (see ref. 2 for a recent proposal). Without such mechanisms, many fundamental questions cannot be answered. For instance, why all previous investigations observed long-range correlations despite their radically different approaches? How and which correlations can flow from the high-level semantic structures down to the crude symbolic sequence in the presence of so many arbitrary influences? What information is gained on the large structures by looking at smaller ones? Finally, what is the origin of the longrange c...

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We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter map is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral analysis of generalized transfer operators. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.

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We consider weakly coupled analytic expanding circle maps on the lattice Z d (for d ≥ 1), with small coupling strength and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fréchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure µ previously obtained by Bricmont-Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For d = 1 we also construct Banach spaces of densities with respect to µ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are O( )-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [BK1].Typeset by A M S-T E X 1 5

We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of "chaoticity". The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the automorphisms, translations and skew translations. We then treat some discontinuous transformations such as the Baker map and the sawtooth-like maps. Our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple.

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open systems. As an example, we explicitly derive the exact size-and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of the closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in many areas of the physical sciences such as information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in general dynamical systems.