Ergodic Theory

Cornfeld, I. P.; Fomin, S. V.; Sinaĭ, Ya. G.

doi:10.1007/978-1-4615-6927-5

Cited by 1,483 publications

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“…This change has a dramatic consequence for the classical ray dynamics inside the billiard, namely the latter becomes fully chaotic. In fact such a dispersive classical billiard is rigorously known to be a K-system [3]. Quite surprisingly, this has also a dramatic effect on the result of the double slit experiment.…”

Section: Chaos Induced Decoherencementioning

confidence: 99%

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Quantum chaos, dynamical stability and decoherence

Casati

Prosen

2005

Braz. J. Phys.

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We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.

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Section: Chaos Induced Decoherencementioning

confidence: 99%

“…any temporal behavior is a discrete superposition of finitely or countably many Fourier components with discrete frequencies. In the ergodic theory of classical dynamical systems, such a quasi-periodic dynamics corresponds to the limiting case of integrable or ordered motion while chaotic motion requires continuous Fourier spectrum [3].…”

Section: Introductionmentioning

confidence: 99%

Quantum chaos, dynamical stability and decoherence

Casati

Prosen

2005

Braz. J. Phys.

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“…A generalization of the discretized generalized baker map is the following [36]. Suppose f is an automorphism of the finite-measure space (X, A, µ), i.e., f is a one-to-one map of X onto itself such that both f and f −1 are µ-invariant.…”

Section: Chaos-based Cryptography On Integer Numbers and Finite Fieldsmentioning

confidence: 99%

Lessons Learnt from the Cryptanalysis of Chaos-Based Ciphers

Álvarez

Amigó

Arroyo

et al. 2011

Studies in Computational Intelligence

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The idea of using chaotic transformations in cryptography is explicit in the foundational papers of Shannon on secrecy systems (e.g., [96]). Although the word "chaos" was not minted till the 1970s [71], Shannon clearly refers to this very concept when he proposes the construction of secure ciphers by means of measure-preserving, mixing maps which depend 'sensitively' on their parameters. The implementation of Shannon's intuitions had to wait till the development of Chaos Theory in the 1980s. Indeed, it was around 1990 when the first chaos-based ciphers were proposed (e.g., [78], [46]). Moreover, in 1990 chaos synchronization [91] entered the scene and shortly thereafter, the first applications to secure communications followed [56,37]. The idea is remarkably simple: mask the message with a chaotic signal and use synchronization at the receiver to filter out the chaotic signal. The realization though had to overcome the desynchronization induced by the message itself. After this initial stage, the number of proposals which exploited the properties of chaotic maps for cryptographical purposes, grew in a spectacular way.

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“…§4 of Chapter 10 of [3]). That is, it is known that there exists an ergodic automorphism T of a separable nonatomic probability measure space ( Ω, A, µ), and a measure preserving transformation S from…”

Section: Theorem 1 (Cf Theorem 1 On P 62 Of [6]) Let ϕ : R → [0 ∞mentioning

confidence: 99%