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A Theory of Program Size Formally Identical to Information Theory
, 1975
"... A new definition of programsize complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest selfdelimiting program for calculating strings A and B if one is given a minimalsize selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) ..."
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Cited by 331 (16 self)
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A new definition of programsize complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest selfdelimiting program for calculating strings A and B if one is given a minimalsize selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be selfdelimiting, i.e. no program is a prefix of another, and (2) instead of being given C and D directly, one is given a program for calculating them that is minimal in size. Unlike previous definitions, this one has precisely the formal 2 G. J. Chaitin properties of the entropy concept of information theory. For example, H(A;B) = H(A) + H(B=A) + O(1). Also, if a program of length k is assigned measure 2 \Gammak , then H(A) = \Gamma log 2 (the probability that the standard universal computer will calculate A) +O(1). Key Words and Phrases: computational complexity, entropy, information theory, instantaneous code, Kraft inequality, minimal program, probab...
Nonstandard smooth realization of translations on the torus. preprint
"... Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action preserving a positive smooth volume. On M, we construct volumepreserving diffeomorphisms that are metrically isomorphic to ergodic translations on the torus, translations in which one given coordin ..."
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Cited by 3 (3 self)
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Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action preserving a positive smooth volume. On M, we construct volumepreserving diffeomorphisms that are metrically isomorphic to ergodic translations on the torus, translations in which one given coordinate of the translation is an arbitrary Liouville number. To obtain this result, we explicitly construct the sequence of successive conjugacies in AnosovKatok’s periodic approximation method, with suitable estimates of their norm. To visualize the construction, we include numerous graphics. 1
Random walk in random scenery: A survey of some recent results
, 2006
"... Abstract. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On Z d, d ≥ 1, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process w ..."
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Cited by 1 (0 self)
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Abstract. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On Z d, d ≥ 1, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by Z, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random walk (and discuss extensions to stationary walk increments and stationary scenery components as well). We describe a number of results for scenery reconstruction and close by listing some open questions. 1.
Nonstandard couples of angles of rotations
, 2012
"... Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action S t preserving a positive smooth volume. We show that on M, the smooth closure of the smooth volumepreserving conjugation class of some Liouville rotations S α of angle α contains a smooth volumep ..."
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Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action S t preserving a positive smooth volume. We show that on M, the smooth closure of the smooth volumepreserving conjugation class of some Liouville rotations S α of angle α contains a smooth volumepreserving diffeomorphism T that is metrically isomorphic to an irrational rotation Rβ on the circle, with α � ±β, and with α and β chosen either rationally dependent or rationally independent. In particular, if M is the closed annulus [0, 1] × � 1, M admits a smooth ergodic pseudorotation T of angle α that is metrically isomorphic to the rotation Rβ. Moreover, T is smoothly tangent to S α on the boundary of M. 1
A smooth GaussianKronecker diffeomorphism
, 2013
"... We construct a smooth GaussianKronecker diffeomorphism T, on � × [0, 1] �, where [0, 1] � is the Hilbert cube. To obtain this diffeomorphism, we adapt a construction by De La Rue [6], which uses transformations of the planar Brownian motion. hal00869084, version 1 2 Oct 2013 1 ..."
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We construct a smooth GaussianKronecker diffeomorphism T, on � × [0, 1] �, where [0, 1] � is the Hilbert cube. To obtain this diffeomorphism, we adapt a construction by De La Rue [6], which uses transformations of the planar Brownian motion. hal00869084, version 1 2 Oct 2013 1
Complemented sets, difference sets, and weakly wandering sequences
, 2007
"... We consider the descriptive complexity of several sets of sequences of natural numbers, and show that the following are all complete analytic sets: the set of complemented sequences, the set of sequences containing an infinite difference set, the set of sequences which are weakly wandering sequences ..."
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We consider the descriptive complexity of several sets of sequences of natural numbers, and show that the following are all complete analytic sets: the set of complemented sequences, the set of sequences containing an infinite difference set, the set of sequences which are weakly wandering sequences for some transformation, and several variants of these. We then use the same techniques to produce weakly wandering sequences with special properties, such as a sequence which is exhaustive weakly wandering for some transformation but which is not weakly wandering for any ergodic transformation. In this paper we consider descriptive aspects of weakly wandering sequences. These sequences are isomorphism invariants for measurepreserving transformations or Borel automorphisms introduced by Hajian and Kakutani in [6]. We first consider how difficult it is to determine whether some sequence can be a weakly wandering sequence for some transformation, an