967 research outputs found
Minimal complexity of equidistributed infinite permutations
An infinite permutation is a linear ordering of the set of natural numbers.
An infinite permutation can be defined by a sequence of real numbers where only
the order of elements is taken into account. In the paper we investigate a new
class of {\it equidistributed} infinite permutations, that is, infinite
permutations which can be defined by equidistributed sequences. Similarly to
infinite words, a complexity of an infinite permutation is defined as a
function counting the number of its subpermutations of length . For infinite
words, a classical result of Morse and Hedlund, 1938, states that if the
complexity of an infinite word satisfies for some , then the
word is ultimately periodic. Hence minimal complexity of aperiodic words is
equal to , and words with such complexity are called Sturmian. For
infinite permutations this does not hold: There exist aperiodic permutations
with complexity functions growing arbitrarily slowly, and hence there are no
permutations of minimal complexity. We show that, unlike for permutations in
general, the minimal complexity of an equidistributed permutation is
. The class of equidistributed permutations of minimal
complexity coincides with the class of so-called Sturmian permutations,
directly related to Sturmian words.Comment: An old (weaker) version of the paper was presented at DLT 2015. The
current version is submitted to a journa
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
Permutation Complexity via Duality between Values and Orderings
We study the permutation complexity of finite-state stationary stochastic
processes based on a duality between values and orderings between values.
First, we establish a duality between the set of all words of a fixed length
and the set of all permutations of the same length. Second, on this basis, we
give an elementary alternative proof of the equality between the permutation
entropy rate and the entropy rate for a finite-state stationary stochastic
processes first proved in [Amigo, J.M., Kennel, M. B., Kocarev, L., 2005.
Physica D 210, 77-95]. Third, we show that further information on the
relationship between the structure of values and the structure of orderings for
finite-state stationary stochastic processes beyond the entropy rate can be
obtained from the established duality. In particular, we prove that the
permutation excess entropy is equal to the excess entropy, which is a measure
of global correlation present in a stationary stochastic process, for
finite-state stationary ergodic Markov processes.Comment: 26 page
Disjointness properties for Cartesian products of weakly mixing systems
For we consider the class JP() of dynamical systems whose every
ergodic joining with a Cartesian product of weakly mixing automorphisms
() can be represented as the independent extension of a joining of the
system with only coordinate factors. For we show that, whenever
the maximal spectral type of a weakly mixing automorphism is singular with
respect to the convolution of any continuous measures, i.e. has the
so-called convolution singularity property of order , then belongs to
JP(). To provide examples of such automorphisms, we exploit spectral
simplicity on symmetric Fock spaces. This also allows us to show that for any
the class JP() is essentially larger than JP(). Moreover, we
show that all members of JP() are disjoint from ergodic automorphisms
generated by infinitely divisible stationary processes.Comment: 24 pages, corrected versio
Extensive amenability and an application to interval exchanges
Extensive amenability is a property of group actions which has recently been
used as a tool to prove amenability of groups. We study this property and prove
that it is preserved under a very general construction of semidirect products.
As an application, we establish the amenability of all subgroups of the group
IET of interval exchange transformations that have angular components of
rational rank~.
In addition, we obtain a reformulation of extensive amenability in terms of
inverted orbits and use it to present a purely probabilistic proof that
recurrent actions are extensively amenable. Finally, we study the triviality of
the Poisson boundary for random walks on IET and show that there are subgroups
admitting no finitely supported measure with trivial boundary.Comment: 28 page
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