Given any K > 0, we construct two equivalent C2 flows, one of which has positive topological entropy larger than K and admits zero as the exponential growth of periodic orbits, in contrast, the other has zero topological entropy and super-exponential growth of periodic orbits. Moreover we establish a C∞ flow on S2 with super-exponential growth of periodic orbits, which is also equivalent to ano...

We study a quantum phase transition between a phase which is topologically ordered and one which is not. We focus on a spin model, an extension of the toric code, for which we obtain the exact ground state for all values of the coupling constant that takes the system across the phase transition. We compute the entanglement and the topological entropy of the system as a function of this coupling...

In this paper I will provide an introduction to the theory of cellular automata and review a paper by Michele D’amico, Giovanni Manzini, and Luciano Margara [3], On computing the entropy of cellular automata, in which the authors study the topological entropy of cellular automata. The main problems addressed are proving a closed form for the topological entropy of D-dimensional linear cellular ...

For a compact metric space X let G = H(X) denote the group of self homeomorphisms with the topology of uniform convergence. The group G acts on itself by conjugation and we say that X satisfies the topological Rohlin property if this action has dense orbits. We show that the Hilbert cube, the Cantor set and, with a slight modification, also even dimensional spheres, satisfy this property. We al...

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal densit...

Entropy is a general concept that appears in different settings with different meanings. Thus, it measures disorder in physics, uncertainty in information theory, minimum code length in coding theory, (pseudo-)randomness in measure-preserving dynamical systems, complexity in topological dynamics, and algorithmic complexity in computer science. As for its importance, let us remind that it enters...

Recently there has been an extensive study on Relative Semi-Dynamical Systems (RSD-Systems). In this paper, we explore some topological properties of RSD-systems. Here, in particular, minimal RSD-systems are characterized and transitive homeomorphisms are investigated. Moreover, α-level relative topological entropy is extended to RSD-systems. Finally, as a computational example, we develop an R...

A unitary operator U acting on the weakly compact unit ball B of a Hilbert space H gives a topological dynamical system (B, V). We prove in this note that for every ergodic u-invariant measure m, the dynamical system (B, U, m) is conjugated to a strictly ergodic group translation. Especially, the topological entropy of (B, U) is zero. As an application, it follows that any flow on the unit ball...

We calculate exactly the von Neumann and topological entropies of the toric code as a function of system size and temperature. We do so for systems with infinite energy scale separation between magnetic and electric excitations, so that the magnetic closed loop structure is fully preserved while the electric loop structure is tampered with by thermally excited electric charges. We find that the...