1 Introduction
In the works [Br74], [Br82] A. Brudno have showed that the entropy of an action is intimately connected with the Kolmogorov complexity of its points. Namely, he proved that the mean Kolmogorov complexity of any poin in the subshift is bounded from above by the topological entropy of the shift, and that there is a point whose mean complexity equals the entropy of the subshift. On the measuretheoretic hand, he proved that, for an ergodic invariant measure on a shiftaction, almost every point has the mean complexity equal to the KolmogorovSinai entropy of the measure.
Later on, the topological part of these results were generalized to the case of a action by Simpson in [Si15]. Alexander Shen brought my attention to the questions in the latter work concerning generalization of Brudno’s results to the case of an arbitrary amenable group.
In my 2013 diploma work [A13] I’ve managed to answer some questions from Simpson’s paper. It is natural to require the group to be computable. Elements of the Følner sequene with high complexity can complicate our matters. So we explicitly require the sequence to satisfy the “modesty” assumption: complexity of a Følner set is negligible relative to its size. We note that there are such Følner sequences in any computable amenable group. Moreover, theorem 6 presents a very natural sufficient geometric criterion for a Følner sequence of a finitelygenerated group to be modest. If elements of a Følner sequence are edgeconnected on the Cayley graph and all contain the group identity element, then the sequence is modest. The following theorem consitutes the main result of my master’s thesis [A13]. That work is written in Russian and have never been published, so I’ve decided to incorporate its results into this paper. Let be an amenable group with a Følner sequence . Let be a finite set. For any we denote
the upper asympotic complexity of relative to , and
the upper asympotic complexity of relative to . Here stands for the Kolmogorov complexity function.
Theorem 1.
Let be a subshift over a computable amenable group. Let be a modest Følner sequence. For every holds
If is also tempered, then there is such a point that
If in addition has continuum cardinality, then there is a continuum of such points .
In proposition 6 from section 7 we handle the upper bound for the complexity in terms of the topological entropy. Proposition 5 from section 7 proves that there is a poin with the complexity not smaller than the topological entropy. The last assertion of the theorem above is due to the fact that if the topological entropy is zero, then every poin will do; if it is positive, then there is a measure of maximal entropy, and almost every point satisfies the requirement. Since the entropy is positive, there is a continuum of these points.
Afterward, Moriakov in his papers [Mo15a] and [Mo15b] independently considered the generalizations of Brudno’s results to the case of amenable groups having socalled computable Følner monotiling. The latter means that the group has a special Følner sequence each of whose elemements can tile the whole group, and that this tiling can be generated by a computer program. This requirement is known to be rather restrictive.
Motivated by Moriakov’s work, I devised a proof for the general case of measureentropy Brudno’s theorem:
Theorem 2.
Let be a computable amenable group. Let be a modest tempered Følner sequence Let be an ergodic inariant measure on . Then for a.e. we have
Proof naturally splits up into lower and upper bounds. The lower bound for the complexity is proved in the proposition 2 from section 4. The upper bound constitutes proposition 4 from section 5. The weaker “mean” form of the latter proposition was used by Bernshteyn in [Be16].
The results of this paper have been announced in [A18].
Acknowledgements. Research is supported by the Russian Science Foundation grant No142100035. I would like to thank Alexander Shen for suggestion to work on generalizations of Brudno’s results, and for helpful discussions. Discussions with Pavel Galashin were instrumental in proving theorem 6. I’d like to thank Sergey Kryzhevich for his comments on my diploma work [A13]; results from that work constitute a substantial part of this paper.
2 Preliminaries
2.1 Computability and Kolmogorov complexity
In the sequel we may assume without loss of generality that our notion of computability is augmented with a fixed oracle. Denote the set of all finite binary strings (the empty one included). For we denote the length of . Let be a computable function (not necessarily everywhere defined). For we denote the smallest such that ( if there are no such ’s). By the KolmogorovSolomonoff theorem, there is a computable function such that for any other computable there is a constant such that for any holds
We fix any such “optimal” . The Kologorov complexity of a string is defined as and denoted by . It is easy to see that for any natural there are at most words satusfying the bound . It is worth to note that we can define the Kolmogorov complexity as a function on any class of finite constructible combinatorial objects. One important example: there is a constant K such that for any finite set and any holds
Another example is complexity of tuples. For a fixed natural there is such that for any tuple holds
2.2 Computable amenable groups
Let be a countable group. Let be a sequence of finite subsets of . The sequence is called a Følner sequence for if for every holds
A countable group is called an amenable group if it has a Følner sequence.
Let be a countable group. We will say that is a computable group if the set of its elements is the set of natural numbers, and the composition function is computable. We may asssume also that is the identity element. We sometimes will use the order inherited from the set of natural numbers for compuable groups. Note that a finitelygenerated group is computable whenever the word problem for the group is decidable. A computable amenable group is simply a computable group which is amenable. As was mentioned earlier, we can define the Kolmogorov complexity of a finite subset of . We will say that a Følner sequence of a computable amenable group is modest if the following bound holds:
This definition was given in my work [A12], the following was also proved in that work.
Theorem 3.
Every computable amenable group has a modest Følner sequence.
Sketch of proof.
Let us fix an enumeration of finite subset of . For let be the first subset such that and that for every (the order is the order of natural numbers). It is easy to see that this sequence is Følner and modest. ∎
In section 3 we will introduce a nice geometric criterion for a Følner sequence to be modest, which works in a lot of situations.
2.3 Actions and ergodic theory
We refer the reader to books [Gl03] and [EW11] for the introduction to ergodic theory. Results amenable to the ergodic theory of amenable groups could be found in [Oll85], [OW87],[L01], [KL16].
Let be a countable group, let be a finite alphabet. The shiftaction of group on the space is defined by
for every and . We endow the space with the product topology. It is easy to note that the action defined is an action by homeomorphisms. A map is said to be equivariant if for every and . A subshift is any closed invariant subset of . We can also mean by it the corresponding restriction of the action.
We denote the set of all partial maps from to with finite domains. We denote the domain of for . We denote the word
where , and ’s are listed in the increasing order.
Let be a countable group. By an action of on a standard probabity space we will always mean a measurable and measurepreserving action. Every invariant measure on defines an action of on .
Suppose a countable group
acts on two standard probability spaces
and . A measurepreserving map is said to be a factor map if for a.e. . It is an isomorphism if it is onetoone on a set of full measure.Suppose a countable group acts on a standard probability space . Let be a measurable map to a finite set . We can extend this map to by the equality
Let be a standard Borel space or standard probability space. A partition of is a finite or countable collection of subsets of whose union is the whole set. For two partitions and we denote the partition .
Consider an action of a countable group on a standard probability space . For a partition of and an element we denote the partition . Partition is said to be generating if the smallest subalgebra containing for each is (modulo sets of mesure ) the algebra of all measurable subsets of . The following condition is wellknown to be equivalent: there is a subset of full measure such that for every pair with there is an element such that and belong to different pieces of .
A probability vector is a countable or finite sequence of nonnegative numbers whose sum is
. Let be a probability vector. It’s Shannon entropy is defined aswith the convention , and denoted by . Any partition of a probability space defines a probability vector. For a partition we denote the entropy of the corresponding probability vector.
Let a countable amenable group act on a standard probability space , let be a partition of finite Shannon entropy. Let be a Følner sequence for . We denote
It is well known that this limit does not depend on the choice of the Følner sequence. The KolmogorovSinai entropy of the action is defined by the formula
Consider the shiftaction of a countable group on . Preimages of elements of under the map from to constitute a partition. This partition is called a canonical alphabet generating partition.
Let a countable group act on a standard probability space . Suppose is a map from to a finite set . Let be the corresponding partition of (namely, the partition consisting of preimages under of elements in ). Suppose is a generating partition. It is well known that the extended map is a bijection on a subset of full measure. We endow now with the pushforward measure . It follows that is an isomorphism of actions. Let be an inverse of , we can assume that this map is Borel. So is an isomorphism from to . This map is an isomorphism of actions also. Alltogether we constructed a symbolic representation, corresponding to partition , of the action of on .
An action of a countable group on a standard probability space is called essentially free if it is free on the orbit of almost every point.
Theorem 4 (Seward and TuckerDrob).
Consider an essentially free action of a countable group on a standard probability space . If , then for every there is a symbolic representation of this action by the shiftaction on , such that the canonical alphabet generating partition has entropy smaller than .
Strictly speaking, the theorem above is a combination of corollary 2.7 from [SeTD16] which states that the socalled Rokhlin entropy is equal to the KolmogorovSinai entropy for free actions of amenable group, and the main result of [Se14] asserting that the statement of the theorem above holds after substituting the KolmogorovSinai entropy with the Rokhlin entropy.
2.4 Topological entropy
For a compact metric space we denote the size of a mininmal spanning net.
Let be a compact metrizable space. Let be a countable amenable group acting by homeomorphisms on . Let us fix a a compatible metric on . Suppose is a finite subset of ; we denote
for . Let us fix a Følner sequence for . Topological entropy of the action is defined by
and denoted as . It is well known that this does not depend on the choice of the metric and Følner sequence.
For a subshift there is the following equivalent definition
We will need the following implication of the variational principle.
Theorem 5.
Let be a finite set and be an amenable group, let be a subshift. There is such an invariant Borel probability measure on that
Proof.
This follows from the variational principle together with the uppersemicontinuity of measure entropy as a function on the set of invariant measures for an expansive dynamical system. See chapter 5 from [Oll85]. ∎
2.5 Asymptotic complexity
Let be a finite alphabet. Consider the shiftaction of a computable amenable group on . Let be a Følner sequence. The upper asymptotic complexity of a point relative to is defined as
and denoted by . The lower asymptotic complexity of a point relative to is defined as
and denoted by .
It is important to note that, if is a modest Følner sequence, then
for every ; analogous equality holds for the lower asymptotic complexity.
3 Reflections concerning modest Følner sequences
The following proposition will be needed to apply the BorelCantelli lemma in the proof of proposition 2.
Proposition 1.
Let be a modest Følner sequence in a computable amenable group . If for every pair with , then for every the following serie converges:
Proof.
Throwing away a finite initial segment of the Følner sequence, we may assume that for every . We rearrange the sum:
For each with we have ; so for each we have . This means that the sum is bounded by the geometric serie:
∎
Theorem 6.
Suppose is a computable amenable group which is finitely generated. Let us fix a finite symmetric generating set . This defines the structure of the Cayley graph on . Let be a Følner sequence. If each is an edgeconnected subset of the Cayley graph and contains the group identity, then is a modest Følner sequence.
Proof.
Algorithm 1 takes a finite connected subset which contains and reversibly encodes it as a binary string consisting of ’s and ’s. This algorithm is based on the standard depthfirst search algorithm for graphs(see [CLRS] chapter 22.3).
I would like to note that the output string of algorithm 1 uniquelly determines the input set. Algorithm 2 decodes the string. I note that stands for the ball of radius in Cayley graph around the group identity. If we apply the encoding to an element of the Følner sequence, we get the binary string of length ; this string contains a negligible amount of ’s; hence, by lemma 1, we have .
∎
4 Measure entropy; lower bound for complexity
Let be a partition of a standard probability space . For denote the element of that contains . We will need the following generalization of the ShannonMcMillanBreiman theorem due to Lindenstrauss [L01]:
Theorem 7.
Consider an ergodic action of a countable amenable group on a standard probability space . Let be a partition of finite Shannon enropy. Let be a tempered Følner sequence. For a.e. holds
.
Proposition 2.
Let be an ergodic invariant measure for the shiftaction of a computable amenable group on . If is a modest tempered Følner sequence, then for a.e. we have
Proof.
Denote . Without loss of generality we may assume that the Følner sequence does not have repeating elements. Let be a canonical alphabet generating partition. Let . By the theorem above, there is a subset with and a number such that for each and we have
In order to prove the proposition it is enough to show that for a.e. we have
For each , denote the subset of all such that . By BorelCantelli lemma, it suffices to show that
We proceed by showing that this is the case. For each there are at most such that . This implies that
So the desired convergence holds due to proposition 1.
∎
5 Measure entropy; upper bound for complexity
Let be an ergodic measure on ; we will show that (under some additional requirements) the upper asymptotic complexity of a.e. point is bounded from above by the KolmogorovSinai entropy. Here’s the general outline of the proof. We show that the asympotic complexity is essentially monotone relative to factor maps. This is done first for cellular maps(which is immediate); then we propagate this to any factormap by means of an approximation argument. We use theorem 4 of Seward and TuckerDrob to show that the desired bound on the complexity holds if the action is essentially free. We relieve the freeness assumption by taking a product with a Bernoulli action of small entropy.
We will use the following ergodic theorem of Lindenstrauss([L01]:
Theorem 8.
Let be a countable amenable group acting ergodically on a standard probability space . Let be an function on . If is a tempered Følner sequence, then
Let be a finite alphabet. For a word we denote the probability vector for occurence rates of letters from in .
The following is lemma 146 from book [SUV].
Lemma 1.
For every holds
Lemma 2.
Consider the shiftaction of a computable amenable group on for a finite set . Let be an ergodic invariant measure. Let be a canonical alphabet generating partition. If is a tempered Følner sequence, then
for a.e. . If in addition is modest, then
for a.e. .
Proof.
The first assertion is a consequence of Lindenstrauss’s ergodic theorem together with the lemma above. The second one follows easily from the observation that for modest Følner sequences and for every holds
∎
Let and be two words in of the same length. The Hamming distance between and is defined as
and denoted by .
The lemma shows two words have close complexity rates given that these words are close in the Hamming metric.
Lemma 3.
For two words , in of the same length the following holds:
Proof.
In order to recover from it is enough to provide the word from that will encode the places where and differ, and the word from that will encode the letters to substitude. The first word contains exactly of ’s. So its complexity is bounded by
The second has complexity bounded by
The statement now follows from standard properties of the Kolmogorov complexity. ∎
For two elements with we define the Hamming distance between them by
and denoted as . Let be a Følner sequence. For two elements we define the asymptotic Hamming distance between them relative to by the formula
and denoted by .
Lemma 4.
Let and be finite sets. Suppose and are two equivariant maps from to . Suppose is an ergodic invariant measure on . Let be the set of all such that , where stands for the identity element in group . Let be a tempered Følner sequence. For a.e. holds
Proof.
It is easy to note that is equivalent to , which in turn is equivalent to . Now the statement follows from Lindenstrauss’s ergodic theorem (theorem 8). ∎
Lemma 5.
In the setting of the previous lemma let be a modest Følner sequence. For a.e. we have
Proof.
This statement is a consequence of the previous lemma together with lemma 3. ∎
Lemma 6.
If is a cellular map, then for every and every Følner sequence holds
Proof.
Suppose is the memory set of cellular map . It is easy to notice that could be uniformly computably recovered from and . Now the statements follows from the definition of a Følner sequence. ∎
Proposition 3.
Let be a measurable equivariant map. Let be an ergodic invariant measure on . For a.e. holds the following bound:
Proof.
Proposition 4.
Let be a finite set. Consider the shiftaction of a computable amenable group on . Let be an invariant ergodic measure on . Let be a modest and tempered Følner sequence. For a.e. holds
Proof.
Denote . Fix any . We will prove that for a.e. holds
Let be a Bernoulli action of entropy smaller than . Consider the productaction . This action is ergodic and essentially free. Its entropy is smaller than . By theorem 4, it has a symbolic representation such that for the canonical alphabet generating partition holds . So we have a factormap from to . By lemma 2, we have for a.e. . By proposition 3 we have that for a.e. holds . This implies that for a.e. . ∎
6 Topological entropy; lower bound for complexity
Proposition 5.
Let be a finite alphabet and let be a computable amenable group. Let be a subshift. If is a modest Følner sequence in , then there is a point such that
7 Topological entropy; upper bound for complexity
Lemma 7.
Let be a computable amenable group.Let be a Følner sequence. For every rational there is a finite sequence and a number such that the following holds. There is a program wich receives a finite subset . If for some , then it outputs the sequence of subsets satisfying the assertions below; otherwise its behaviour is undefined.

,

,

.
Proof.
Proposition 6.
Let be a computable amenable group. Let be a finite alphabet. Suppose is a subshift. Let be a modest Følner sequence for group . For every holds
Proof.
Denote . Take arbitrary rational . It suffices to prove that for every holds
Throwing away a finite initial part of , we may assume that
We apply the previous lemma. This lemma gives us a collection . We also record the finite subsets for the future use. Now we will consruct a program that will take a finite subset of , and if for some , it will output the set of all satisfying for every and (note that is equivalent to ). It is easy to see that for we have
Indeed, denote
it follows from the previous lemma that
Bound (7) follows since
Note that for we have . We also have . The properties of the Kolmogorov complexity imply that for every and for , holds
Hence, (7) holds for every . ∎
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