ON THE COMPUTATION OF ARNOLD COMPLEXITY OF LENGTH 2n BINARY WORDS

Among all binary words w of length 2n, n ≥ 1, we find the words whose Arnold complexity can be calculated by counting the number of occurrences of ones in certain disjoint fragments of w.

ON THE COMPUTATIONAL COMPLEXITY OF THE ARNOLD COMPLEXITY OF BINARY WORDS

For an arbitrary binary word ω of Arnold complexity A(ω) we prove that the number of steps in computing A(ω) can be lowered from A(ω) to ν(A(ω)) where ν(A(ω)) is the number of occurrences of the digit 1 in A(ω). For a class of words we obtain an explicit algorithm for computing A(ω). We illustrate the efficacy of our construction by determining the Arnold complexity of the Thue–Morse sequence.