We further continue the discussion of topological entropy. Here, we discuss various results that bound above and below the entropy of a given map.

1. The topological entropy of Lipschitz maps

Many of the dynamical systems of interest are actually given by compact manifolds {M} and smooth maps {T: M  \rightarrow M}. These are always Lipschitz with respect to appropriate metrics. Indeed, choose a Riemannian metric on {M} and let {||\cdot||} denote the induced norm on the tangent spaces. Then {\sup  ||DT||} is a Lipschitz constant for {T} with respect to the metric on {M} induced by the Riemannian metric. In this case, the entropy is always finite. We shall prove this in a more general context.

Let {X} be a compact metric space, and for {\epsilon>0}, let {b(\epsilon)} denote the the number of {\epsilon}-balls necessary to cover {X} (which is always finite). Then we call

\displaystyle   \limsup_{\epsilon \rightarrow 0} \frac{ \log b(\epsilon) }{ |\log  \epsilon |}

the ball dimension of {X}. For instance, an {n}-cube has ball dimension {n}. It follows more generally that a Riemannian {n}-manifold has ball dimension {n}.

The reason we shall use this concept below is that {b(\epsilon)} gives an upper bound for a minimal {\epsilon}-spanning set of the space {X}. (Recall that an {\epsilon}-spanning set means that every point is {\epsilon}-close to it.) In fact, if the {\epsilon}-balls {B_1, \dots, B_q} cover {X}, then the centers of these form an {\epsilon}-spanning set.

Theorem 1 Let {X} be a compact metric space with finite ball dimension {D}. Suppose {T:X \rightarrow X} is a Lipschitz continuous mapping with constant {L}, i.e. {d(Tx,Ty) \leq Ld(x,y)} for all {x,y  \in X}. Then the entropy of {T} is finite and\displaystyle  h_{top}(T) \leq  \max(0, D \log L). (more…)

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We continue the discussion of topological entropy started yesterday.

-1. Basic properties-

So, recall that we attached an invariant {h_{top}(T) \in [0,  \infty]} to a transformation {T: X \rightarrow  X} of a compact metric space {X}. We showed that it was purely topological, i.e. invariant under semiconjugacies. However, we have yet to establish its basic properties and compute some examples.

In general, we can’t expect topological entropy to be additive, i.e. {h_{top}(T \circ S) = h_{top}(T) + h_{top}(S)}, even if {T} and {S} commute. The reason is that the identity—or any isometry—has zero entropy, while there are homeomorphisms with nonzero entropy.

However, we do have:

Theorem 1 If {m \in  \mathbb{Z}}, then {h_{top}(T^m) = |m|  h_{top}(T)}.

(Here if {m \geq 0}, this makes sense even for {T} noninvertible.)

We handle the two cases {m>0} and {m=-1} (which together imply the result). In each, we will use the second definition of entropy that we gave in terms of coverings (which historically actually came first). Namely, the definition was

\displaystyle  h_{top}(T) =  \sup_{\mathfrak{A}} \lim_{n} \frac{1}{n} \log \mathcal{N}( \mathfrak{A}  \vee T^{-1}\mathfrak{A} \vee \dots \vee  T^{-n+1}\mathfrak{A}). (more…)

In the theory of dynamical systems, it is of interest to have invariants to tell us when two dynamical systems are qualitatively “different.” Today, I want to talk about one particularly important one: topological entropy.

We will be in the setting of discrete dynamical systems: here a discrete dynamical system is just a pair {(T,X)} for {X} a compact metric space and {T: X  \rightarrow X} a continuous map.

Recall that two such pairs {(T,X), (S,Y)} are called topologically conjugate if there is a homeomorphism {h: X \rightarrow Y} such that {T =  h^{-1}Sh}. This is a natural enough definition, and it is clearly an equivalence relation. For instance, it follows that there is a one-to-one correspondence between the orbits of {T} and those of {S}. In particular, if {T} has a fixed point, so does {S}. Admittedly this necessary criterion for determining whether {T,S} are topologically conjugate is rather trivial.

Note incidentally that topological conjugacy needs to be considered even when one is studying smooth dynamical systems—in many cases, one can construct a homeomorphism {h} as above but not a diffeomorphism. This is the case in the Hartman-Grobman theorem, which states that if {f: M \rightarrow M} is a smooth map with a fixed point where the derivative is a hyperbolic endomorphism of the tangent space, then it is locally conjugate to the derivative (that is, the corresponding linear map). (more…)