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2 results

  • Covering Finite Sets by Ergodic Images

  • J. Michael Steele
  • Journal:
    Canadian Mathematical Bulletin / Volume 21 / Issue 1 / 01 March 1978
    Published online by Cambridge University Press:
    20 November 2018, pp. 85-91
    Print publication:
    01 March 1978
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  • For any ergodic transformation T a set A of measure less than ∈ is constructed with the property that for every finite set F there is a j = j(F) such that F ⊂ T-iA. The basic tool used to prove this is a purely combinatorial result which says there is a small subset of { l, 2, …, n } which can be shifted a small amount to cover any k set in {j: δn ≤j≤n}. Applications are given to the theory of combinatorial entropy.

  • Positive columns for stochastic matrices

  • Dean Isaacson, Richard Madsen
  • Journal:
    Journal of Applied Probability / Volume 11 / Issue 4 / December 1974
    Published online by Cambridge University Press:
    14 July 2016, pp. 829-835
    Print publication:
    December 1974

  • If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.