Ergodic-Type Characterizations of Algorithmic Randomness
نویسندگان
چکیده
A theorem of Kučera states that given a Martin-Löf random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A. We show that this result can be seen as an effective version of Birkhoff’s ergodic theorem (in a special case). We prove several results in the same spirit and generalize them via an effective ergodic theorem for bijective ergodic maps.
منابع مشابه
On Stability Property of Probability Laws with Respect to Small Violations of Algorithmic Randomness
We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff’s pointwise ergodic theorem for ergodic transformations, are stable in this...
متن کاملMultiple Recurrence and Algorithmic Randomness
This work contributes to the programme of studying effective versions of “almost everywhere” theorems in analysis and ergodic theory via algorithmic randomness. We determine the level of randomness needed for a point in a Cantor space {0, 1} with the uniform measure and the usual shift so that effective versions of the multiple recurrence theorem of Furstenberg holds for iterations starting at ...
متن کامل. L O ] 1 2 Ju n 20 12 RANDOMNESS AND NON - ERGODIC SYSTEMS
We characterize the points that satisfy Birkhoff’s ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Löf random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the...
متن کاملComputability of the ergodic decomposition
The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-...
متن کاملAn algorithmic view of pseudochaos
The controversial concept of quantum chaos – the dynamical chaos in bounded mesoscopic quantum systems – is presented as the most important and universal instance of a new generic dynamical phenomenon: pseudochaos. The latter characterizes the irregular behaviour of dynamical systems with discrete energy and/or frequency spectrum, which include classical systems with discrete phase space. The q...
متن کامل