Symbolic dynamics and hyperbolic dynamic systems

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

“…Moreover, s is the optimal expected asymptotic complexity rate, in the sense that every sequence of projectors q n ∈ A (n) , n ∈ N, that for large n may be represented as a sum of mutually orthogonal one-dimensional projectors that all violate the lower bounds in (2) and (3) for some δ > 0, has an asymptotically vanishing expectation value with respect to Ψ .…”

“…U(p) = i (n) . For infinite sequences i, in analogy with the entropy rate, one defines the complexity rate as k(i) := lim n 1 n K(i (n) ), where i (n) is the string consisting of the first n bits of i, [2]. The universality of U implies that changing the UTM, the difference in the complexity of a given string is bounded by a constant independent of the string; it follows that the complexity rate k(i) is UTM-independent.…”

Are chaotic systems dynamically random?

“…Therefore, if one assumes equidistribution, the initial configuration of a classical system must be represented by an uncomputable number with probability one. Since present-day definitions of uncomputability are essentially equivalent to (normalized) randomness [2,3], this renders chaotic motion for deterministic evolution functions capable of "unfolding" the randomness of the initial values. A sufficient criterion for such an evolution function is the instability of trajectories towards variations of the initial configuration δX 0 , such that at later times t and for positive Lyapunov exponent λ + , δX t ≈ δX 0 exp(λ + t).…”

“…The formal definitions will be given next. They rest upon the representation of an experimental sequence in a symbolic string x [2].…”

“…For physical applications, normalized ACR, henceforth called CR randomness, is very important [2]. An infinite sequence x is CR random, if K(x) = lim n→∞ H(x(n))/n > 0.…”