Ergodic measures with infinite entropy
We construct ergodic probability measures with infinite metric entropy for typical continuous maps and homeomorphisms on compact manifolds. We also construct sequences of such measures that converge to a zero-entropy measure.
Let be a compact manifold with finite dimension , with or without boundary. Let be the space of continuous maps with the uniform norm:
We denote by the space of homeomorphisms with the uniform norm:
A subset (or ) is called a -set if it is the countable intersection of open subsets of (resp. ). We say that a property of the maps (or ) is typical, or that typical maps satisfy , if the set of maps that satisfy contains a dense -set in (resp. ).
The main result of this article is the following theorem.
The typical map has an ergodic Borel probability measure such that
In the case that is a compact interval, Theorem 1 was proved in [CT]. Yano proved that typical continuous maps of compact manifolds with or without boundary have infinite topological entropy [Ya]. Therefore, from the variational principle, there exists invariant measures with metric entropies as large as wanted. Nevertheless, this property alone does not imply the existence of invariant measures with infinite metric entropy. In fact, it is well known that the metric entropy function is not upper semi-continuous for -typical systems. Moreover, we prove that it is strongly non upper semi-continuous in the following sense:
For a typical map there exists a sequence of ergodic measures such that
where denotes the limit in the space of probability measures endowed with the weak topology.
Even if it were a-priori known that some -invariant measure has infinite metric entropy, this property alone would not imply the existence of ergodic measures with infinite metric entropy as Theorems 1 and 2 state. Actually, if had infinitely many ergodic components, to prove that the metric entropy of at least one of its ergodic components must be larger or equal than the entropy of , one needs again the upper semi-continuity of the metric entropy function (see for instance [Ke, Theorem 4.3.7, p. 75]).
Yano also proved that typical homeomphisms on manifolds of dimension 2 or larger, have infinite topological entropy [Ya]. Thus one wonders if Theorems 1 and 2 hold also for homeomorphisms. We give a positive answer to this question.
If the dimension of the manifold is at least 2, then the typical homeormorphism has an ergodic Borel probability measure such that
If the dimension of the manifold is at least 2, then for a typical homeomorphism there exists a sequence of ergodic measures such that
1.1. Open questions
If is Lipschiz then no invariant measure has infinite entropy, since its topological entropy is finite. The following question arises: do Theorems 1 and 3 hold also for maps with more regularity than continuity but lower regularity than Lipschiz? For instance, do they hold for Hölder-continuous maps?
“A-priori” there is a chance to answer positively this question for one-dimensional Hölder continuous endomorphisms, because in such a case, the topological entropy is typically infinite [Ha]. Also for bi-Hölder homeomorphisms on manifolds of dimension 2 or larger, there is a chance to answer positively the above question, because their topological entropy is also typically infinite [FHT], [FHT1]. In this article we will focus only on the -case, and leave for further research the eventual adaptation of our proofs, if this adaptation is possible, to -maps for homeomorphisms with .
The hypothesis of Theorems 1 and 3 states that is a compact manifold. It arises the following question: do some of the results also hold in other compact metric spaces that are not manifolds? For instance, do they hold if the space is a Cantor set ?
If the aim were just to construct with ergodic measures with infinite metric entropy, the answer would be positive. But if the purpose were to prove that such homeomorphisms are typical in , the answer would be negative.
In fact, Theorem 3 holds in particular for the 2-dimensional square . One of the steps of the proofs consists in constructing some fixed Cantor set , and a homeomorphism on that leaves invariant, and possesses an -invariant ergodic measure supported on with infinite metric entropy (see Lemma 3.1 and Remark 3.2). Since any pair of Cantor sets and are homeomorphic, we deduce that any Cantor set supports a homeomorphism and an -ergodic measure with infinite metric entropy.
Nevertheless, the above phenomenon is not typical on a Cantor set . On the one hand, there also exists homeomorphisms on with finite, and even zero, topological entropy. (Take for instance conjugated to the homeomorphism on the attractor of a Smale horseshoe, or to the attractor of the - Denjoy example on the circle.) On the other hand, it is known that each homeomorphism on a Cantor set is topologically locally unique; i.e., it is conjugated to any of its small perturbations [AGW]. Therefore, the topological entropy is locally constant in . We conclude that the homeomorphisms on the Cantor set with infinite metric entropy, that do exist, are not dense in ; hence they are not typical.
1.2. Organisation of the article.
We construct a family , called models, of continuous maps in the cube , including some homeomorphisms of the cube onto itself if , which have complicated behavior on a Cantor set (Definition 2.6). A periodic shrinking box is a compact set that is homeomorphic to the cube and such that for some : are pairwise disjoint and (Definition 4.1). The main steps of the proofs of Theorems 1 and 3 are the following results.
Lemma 3.1 Any model in the cube has an -ergodic measure such that
A good sequence of periodic shrinking boxes is a sequence of periodic shrinking boxes such that accumulate (with the Hausdorff distance) on a periodic point , and besides their iterates also accumulate on the periodic orbit of , uniformly for (see Definition 5.1). The main step in the proof of Theorems 2 and 4 is Lemma 3.1 together with
Lemma 5.2 Typical maps , and if also typical homeomorphisms, have a good sequence of boxes, such that the return maps are topologically conjugated to models .
2. Construction of the family of models.
We call a compact set or more generally where is an -dimensional manifold a box if it is homeomorphic to . Models are continuous maps of the . We will discuss separately the cases and .
(Models in the interval.)
Let . We call a model if and there exists an ergodic -invariant measure such that .
[CT, Theorem 39](Existence of models )
The family of models in the interval is a dense -set in .
2.1. Models in dimension 2 or larger
In this subsection, we assume . We denote by the space of relative homeomorphisms (i.e., is a homeomorphism onto its image included in ), with the topology induced by:
(-relation from a box to another).
Let . Let be two boxes. We write
Observe that this condition is open in and also in .
(Atoms of generation 0 and 1) (See Figure 2)
We call a box an atom of generation 0 for , if there exists two disjoint boxes , which we call atoms of generation 1 such that
If is an atom of generation 0 and are the two atoms of generation 1, we denote ,
Atoms of generation (See Figure 2)
Assume by induction that the finite families of atoms for of generations up to are already defined, such that the atoms of the same generation are pairwise disjoint, contained in the interior of the -atoms in such a way that all the -atoms contain the same number of -atoms and
Assume also that for all and for all :
We call the boxes of a finite collection of pairwise disjoint boxes, atoms of generation , or -atoms if they satisfy the following conditions (see Figure 2):
Each atom of generation is contained in the interior of an atom of generation , and the interior of each atom contains exactly pairwise disjoint -atoms, which we call the children of . We denote
For each pair , the collection of children of is partitioned in sub-collections as follows:
For each 3-uple there exists exactly -atoms, say such that
For example, in Figure 2 we have .
Besides, we assume the following properties for all
From the above conditions we deduce:
and for any pair :
We also deduce the following properties for any atom :
In fact, on the one hand for each atom , the number of atoms such that equals the number , where and are the unique -atoms such that for some . On the other hand, the number of atoms such that equals
where and are the unique -atoms such that .
Since if , we conclude that not all the pairs of atoms of generation satisfy .
Remark. The above conditions are open in and in .
(Models in dimension 2 or larger)
Denote by the family of all the models in .
If , the family is a non-empty -set in and is a non-empty -set in . In fact, as remarked at the end of Definition 2.5, for each the family of maps that have atoms up to generation , is open in and also in . Besides, the conditions and are also open.
2.2. Paths of atoms
Let , and let be a -uple of atoms for of the same generation , such that
We call an -path of -atoms from to . Let denote the family of all the -paths of atoms of generation .
For all , and for all there exists and an -path of -atoms from to .
For , the result is trivial with (see Definition 2.4). Let us assume by induction that the result holds for some and let us prove it for .
Let . From Definition 2.5, there exists unique atoms such that Then and
Analogously, there exists unique atoms such that Then and
Since the induction hypothesis ensures that there exists and an -path from to . We write , . So Assertion (3) becomes
Let . For each -path of -atoms there exists an - path of atoms of generation such that for all
In the proof of Lemma 2.9 for each -path of -atoms we have constructed the -path of -atoms as wanted. ∎
2.3. Construction of models in dimension larger than 1.
Lemma 2.2 states that the family of models is a nonempty -set in . In the case , we will prove the following result, whose difficult part is b). The difficulty would not be so, if we were only constructing models . For further uses in Section 4, we need to be besides isotopic to an arbitrarily given homeomorphism .
If , then
a) The family of models is a nonempty -set in and is a nonempty -set in .
b) For all such that there exists and such that
We will divide the construction of and into several steps:
Step 1. Construction of the atoms of generation 0 and 1.
Since , there exists a box such that . The box is the atom of generation 0 or 0-atom for a map that we construct in this step. Let . We choose two pairwise disjoint boxes and contained in . They are the atoms of generation 1, or for short 1-atoms for . Let . We choose them small enough such that
Let be an -dimensional box () and consider any two finite sets and of points in . A simple proof by induction on shows that there exists a homeomorphism which is the identity on and satisfies . We apply this result to the box as follows. For each we choose two different points . Then there exists a homeomorphism such that
Now, we extend to be a homeomorphism of the whole box by defining . In particular
Nevertheless, we will not use the homeomorphisms and as they are. We will modify them to obtain new homeomorphisms and such that for any ordered 3-uple the interior of the intersection is nonempty. This additional property will allow us to construct later the atoms of generation 2 for a new homeomorphism without changing too much the previous homeomorphism .
We will modify only in the interiors of the boxes to obtain a new homeomorphism such that for all Therefore, the same atom of generation 0 and the same two atoms of generation 1 for , will still be atoms of generations 0 and 1 for the new homeomorphism defined by .
From the construction of and , for any ordered pair there exists a point . Denote by
the connected component of such that . Choose two different points
and a (one-to-one and surjective) permutation of the finite set
for some and some in such a way that for each fixed ordered 3-uple there exists one and only one point such that .
Now, for each we construct a homeomorphism
Such exists as the composition of with a homeomorphism that leaves fixed the points of for each both atoms , and transforms each point on .
Now, we extend to the whole box by defining In particular
As mentioned at the beginning, the property for both atoms implies that they are also atoms of generation 1 for . But now, they have the following additional property:
There exists a one-to-one correspondence between the 3-uples
and the eight points of the set in Equality (7), such that
Step 2. Construction of the atoms of generation .
Assume by induction that we have constructed the families of atoms up to generation for , where is a homeomorphism that leaves fixed the points of , and such that
Assume also the following two assertions for all :
for all there exists a point such that
is -invariant, and
where and are fixed connected components of and of respectively.
Let us construct the family of atoms of generation and the homeomorphisms and . First, for each we choose a box such that
Note that , the set is -invariant, and besides thus
Next, for each we choose two pairwise disjoint boxes contained in the interior of , satisfying
Note that for all ,
The boxes of the family will be the -atoms of a new homeomorphism that we construct as follows.
In the interior of each box we choose different points , and denote
Then, we build a permutation of the set such that for all , and for each box , it transforms the point into the point
for some and for some box