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 for
a compact metric space and
a continuous map.
Recall that two such pairs are called topologically conjugate if there is a homeomorphism
such that
. 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
and those of
. In particular, if
has a fixed point, so does
. Admittedly this necessary criterion for determining whether
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 as above but not a diffeomorphism. This is the case in the Hartman-Grobman theorem, which states that if
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).
1. Definition of topological entropy
Anyway, we need new invariants. One extremely important one is topological entropy, which measures in some sense the “complexity” of . Consider the following problem. For a natural number
, consider segments
for all
. How many of them are there?
Clearly, the answer will be infinite in general. But we can count how many we need to get a dense packing in the space of all such segments. To be precise, for , define the number
to be the minimal natural number
such that there exist points
such that for every
, there is some
such that
Here is the metric on
. The topological entropy
is defined as
This is a rather complex definition, so it will be useful to pause to think again about it. Another way to do this is to introduce a new metric on .
Namely, we define the metric via
. Then, in any metric space
, we can call a subset
-dense if every point of
is of distance
from some point of
. The selection of points
as above was made so that
is an
-dense set—indeed, the smallest such—in
endowed with the metric
. This provides some motivation for the definition.
There is a variation on the idea of -dense: namely,
-separated. This means that any two distinct points in the given subset (which we call
-separated) have distance
. The problem of finding a maximal
-separated set (“to pack the points such that they are far away from each other”) is related to the problem of finding a minimal
-dense set. Namely, one can check that a minimal
-dense set is
-separated, and similarly a maximal
-separated set is
-dense.
This provides another way of thinking of topological entropy. Let denote a maximal
-separated subset of
. Then
2. A more natural definition
I personally find this definition a little strange. For one thing, it appears superficially to depend on the metric , while we supposedly just care about the topological structure. In addition, the formula is rather complicated. We have yet to show that it is invariant under topological conjugacy, in fact.
The original definition of Adler, Konheim, and McAndrew is simpler and seems more natural to me; it is defined very explicitly in terms of coverings. It does not even use the metric structure of . So, fix a compact space
, and let
be a continuous map, as before. Now an open covering will be denoted
. The refinement
of two open coverings
is just the covering
. We define the size
of the cover
to be the cardinality of the minimal subcover; obviously
.
Given an open cover , we define the inverse image
via
; it is clear that
. The following theorem gives another definition of topological entropy (which is how Walters introduces it)
Theorem 1 The topological entropy is the supremum of
over all open covers
.
This result actually follows rather easily from the definitions. Note that the limit actually exists, because if , then the properties of
mentioned imply that
, from which it is a straightforward exercise in analysis that
equals and is the infimum.
Indeed, suppose is the cover by all
-balls. Take the metric
as above. Any set
for
has
-diameter at most
. Then if
is the size of a minimal
-spanning set with respect to the metric
as above, clearly
because taking one point from each set in the covering gives an
-spanning set for
, and this spanning set is minimal if and only if the cover is minimal. It follows that
In particular, the topological entropy is less than or equal to
and is equal to the limit as of that quantity for
the cover by
-balls.
Now, if is any cover, I claim that the limit exists and is at most
. This will prove the theorem. This is because there is a Lebesgue number
for
, and the limit is at most that of the limit with
replaced by the cover of
-balls. So we can reduce to this case, which was already handled above.
Incidentally, it clearly suffices to take finite (because shrinking
only increases the limit). This equation is evidently invariant under topological conjugacy, so the theorem implies:
Corollary 2 Topological entropy is invariant under topological conjugacy.
Next time, we’re going to compute some explicit examples of what this actually means, as well as proving a few more elementary properties.
July 15, 2010 at 5:51 pm
[…] Topological entropy II July 15, 2010 tags: dynamical systems, periodic points, topological entropy by Akhil Mathew We continue the discussion of topological entropy started yesterday. […]
July 17, 2010 at 5:02 pm
Hi, I am not quite sure about the proof of theorem 1, when we take
to be all the
balls. I can only see
In order to get another direction need some more careful analysis, I think.
July 17, 2010 at 6:15 pm
Well, suppose we have
points
such that they
-span
with respect to
. This means that if
, there is
such that
are
-close for all
.
. This family covers
by definition. It is also a family of open sets in
.
Then consider the sets
It was a bit more subtle than I initially thought though.
July 17, 2010 at 6:42 pm
yeah, I get it, thanks.