Conductance, Laplacian and Mixing Rate in Discrete Dynamical Systems

Abstract

We introduce the notion of conductance in discrete dynamical systems defined by iterated maps of the interval. Our starting point is the notion of conductance in the graph theory. We pretend to apply the known results in this new context.

Conductance in discrete dynamical systems

“…Several studies on this new invariant followed, such as [4], [8], [17], and [12]. This new topological invariant was related with conductance in [9] and [6], which arise naturally in the context of reversible Markov chains.…”

“…This kind of trajectory is called eventually periodic. This family corresponds to isentropic maps, see [4], with constant Φ 1 (see [6]) and indicates the need of other invariants. In this case, the use of Φ 2 , Φ 3 or Φ 4 is completely justified as can be seen in Fig.…”

“…We use the context of graph theory (see [3], [14], and [15]) and discrete dynamical systems to establish the analogy between some concepts ( [6] and [5]): edge and vertex expansion versus mixing time, graph isoperimetric numbers versus conductance, and the Laplacian of a graph versus the discrete Laplacian.…”

“…In [6] was introduced the conductance of a discrete dynamical system using the adapted weighted version of i 1 as was done in [3] for the correspondent weighted graph…”

Conductance and Noncommutative Dynamical Systems

“…Similarly, the conductance introduced in physics, particularly in electrical circuits, is now present in graph theory, random walks, knot theory, and it is related to mixing properties and the convergence rate of the system to an equilibrium state, see for example [1][2][3]. In [4], the authors study the connection between the conductance in graph theory and the conductance associated to Markov subshifts, in particular related to the second eigenvalue of the Markov matrix. This allows us to distinguish systems with the same entropy, see also [5].…”