Predictability, topological entropy and invariant random orders
Andrei Alpeev Tom Meyerovitch Sieye Ryu




Three-Dimensional Krawtchouk Descriptors for Protein Local Surface Shape Comparison
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    December 30, 2023
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	We  prove  that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route,  we investigate  invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. 
	We also present a proof  due to  Weiss  for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy. 





The aim of this note is to prove the following:




Let Γ  be a countable amenable group that acts on

a compact metric space X by homeomorphism,  and let S ⊂Γ be a subsemigroup of Γ with 1_Γ∉S.

If every continuous function f ∈ C(X) is contained in the closed algebra generated by {f ∘ s :  s ∈ S} and the constant functions, then the action ΓX has zero topological entropy.



Theorem <ref> was initially proved in the case Γ= and S = _+ by Kaminśki, Siemaszko and Szymanśki in <cit.> (see also <cit.> and  <cit.>). In <cit.> Hochman gave another proof and also generalized this to the case Γ =.

In the same work Hochman conjectured Theorem <ref>. 

Later, Huang, Jin and Ye in <cit.> proved Theorem <ref>  under the additional assumption  that  Γ is torsion-free and locally nilpotent.
It turns out that  the proof of Theorem <ref> does not involve any considerable new ideas or tools beyond those developed by Hochman in the same paper where the question had been posed. However, as suggested to us by L. Bowen, without paying too great a price, we are able to obtain more general results about predictability in the presence of  invariant random orders on groups.
We will deduce Theorem <ref> above as a particular case of the slightly more general Theorem <ref> below.


For a countable group Γ we denote by (Γ) ⊂{0,1}^Γ×Γ the space of all orders on Γ.

The space (Γ) is metrizable and compact, and admits a natural Γ-action.

(see Section <ref>).

Let ν be a Γ-invariant measure on (Γ).
We will say that action ΓX is ν-topologically predictable relative to a topological factor map  π:X → Y if the following holds: For any f ∈ C(X), and ν-a.e. ≺∈(Γ) the function f is contained in the closed algebra generated by { f∘ g: g ≺ 1_Γ} together with the image of  C(Y) in C(X) under the map  π_*:C(Y) → C(X).


	Let Γ  be a countable amenable group that acts on
	
	two compact metric spaces X,Y by homeomorphisms with a   continuous Γ-equivariant map π:X → Y. 
	If there exists
	 a Γ-invariant probability measure ν on the space (Γ)  such that  
	ΓX is ν-topologically predictable relative to  π:X → Y, then the topological entropy of the action ΓX is equal to the topological entropy of the action ΓY.


Theorem <ref> is a special case of Theorem <ref>, obtained by taking ΓY to be the trivial action on a singleton, and taking ν to be the delta measure concentrated on the order ≺_S given by

    g_1 ≺_S g_2  ⇔  g_1g_2^-1∈ S.


In the last decade entropy theory has emerged for actions of non-amenable groups. Entropy for measure preserving  actions of sofic group was developed starting with the seminal paper <cit.>. Sofic topological entropy was introduced by Kerr and Li in <cit.>. The reader may find more details in <cit.>,  <cit.> and <cit.>. It is natural to ask if the above results extend to the non-amenable setting.

In Section <ref> we include a short proof that was communicated to us by Benjy Weiss for the fact that topologically prime actions have zero entropy. Weiss's proof  uses some  similar techniques as the result about predictable systems. The result about prime systems  also applies to actions of sofic groups, with appropriate adjustments. 

Acknowledgements. We thank Benjy Weiss for sharing his argument regarding the entropy of topologically prime systems and Lewis Bowen for insightful comments and references regarding random invariant orders. We also thank Ben Hayes for his comments on the first version of this paper. Andrei Alpeev was supported by “Native towns”, a social investment program of PJSC “Gazprom Neft”. Tom Meyerovitch and Sieye Ryu acknowledge support by the  Israel Science Foundation (grants no. 626/14 and 1052/18) and the The Center For Advanced Studies In Mathematics in Ben Gurion University. Tom Meyerovitch thanks the Pacific Institute for Mathematical Studies and the department of mathematics at the University of British Columbia for warm hospitality.  



§ PRELIMINARIES


In this paper ΓX will denote  a left action of a countable group Γ by homeomorphisms on a compact metric space X.

We will denote by (X) the (compact, convex) space of Borel probability measures on X, and by _Γ(X) the subset of Γ-invariant  Borel probability measures.

We will use the notation Γ(X,μ) to indicate that μ∈_Γ(X), and in this case we will say that the Γ action on (X,μ) is measure preserving.

For two partitions α and β denote their join by α∨β. 
Similarly, for two σ-algebras  and ℬ denote by ∨ℬ the smallest σ-algebra containing both.
We will say that a partition β is finer than a partition α if each element of α is a union of elements from β. 


Let α be a countable measurable  partition of a standard probability space X. We denote H_μ(α) its Shannon entropy; for two partitions α and β we denote H_μ(α|β) the Shannon entropy of partition α relative to partition β. For a partition α and a σ-subalgebra  of a standard probability space, the Shannon entropy of α relative to  is given by

    H_μ(α|) = -∫logμ( α|)dμ.
 
 If α is a partition of finite Shannon entropy and  is a σ-subalgebra, then the following holds:

    H_μ(α| ) = inf{  H_μ(α|β),  where β⊆, H_μ(β)<+∞};

Moreover, if  (β_i)_i=1^∞ is a sequence of partitions having finite Shannon entropy and
 is the smallest σ-subalgebra that contains each β_i then

    H_μ(α|) = lim_i →∞ H_μ(α|⋁_j ≤ iβ_j).




We recall that countable group Γ is  amenable if it has a Følner sequence, namely a sequence (F_i)_i=1^∞ of finite subsets such that for any g ∈Γ the following holds:

    lim_i →∞| g F_i ∖ F_i |/| F_i| = 0.


It is a well known fact that a countable group  Γ is amenable if and only if
any action of Γ by continuous affine transformations on a compact convex subset of a separable locally convex topological vector space has a fixed point (for this and  many other equivalent conditions for amenability, see for instance <cit.>).

 
  For a countable  partition  α = { B_1, B_2, …} and g ∈Γ we denote 

    α^g  = { g^-1(B_1), g^-1(B_2), …}.


If   F ⊂Γ is finite let α^F = ⋁_g ∈ Fα^g.
This is again a countable partition.
If F ⊂Γ is infinite, we let α^F denote 
  the smallest σ-subalgebra containing all α^g for g ∈ F. 
  
  
  Now suppose that (F_i)_i=1^∞ is a Følner sequence for the group Γ.
  
  We will denote the  Kolmogorov-Sinai entropy  of the partition α relative to a Γ-invariant σ-subalgebra ⊂(X) by h_Γ(α, X,μ|). This  is defined by the formula

    h_Γ(α, X,μ|)=
    lim_i →∞H_μ(α^F_i|)/| F_i|.

It is known that h_Γ(α, X,μ|) does not depend on the choice of the Følner sequence. The Kolmogorov-Sinai entropy of the action relative to a σ-subalgebra  will be denoted by h_Γ(X,μ|). This is 
given by

    h_Γ(X,μ|) = sup{ h_Γ(α, X, μ|) :  H_μ(α)<∞}.


We denote  the Kolmogorov-Sinai entropy of the factor corresponding to  by 



    h_Γ(, X,μ) = sup{ h_Γ(α, X, μ) : α⊂, H_μ(α)<∞}.

For an invariant σ-subalgebra  we denote h_Γ(, X, μ|) the entropy of the factor corresponding to  relative to .


The Kolmogorov-Sinai theorem asserts that 

    h_Γ(α^Γ∨ , X, μ|) = h_Γ(α , X, μ|).



The topological entropy of an action ΓX of a countable amenable group Γ on a compact space X will be denoted by h_Γ(X).






§ INVARIANT RANDOM ORDERS AND INVARIANT RANDOM PASTS


In the sequel we will employ some rudimentary theory of random orders on groups. Random orders were successfully used in <cit.> and <cit.> to prove results concerning deterministic orders on amenable groups.
 See the book <cit.> for background and much more. Particular  invariant random orders have been applied to entropy theory, notably Keiffer's paper <cit.> about actions of amenable groups, and  <cit.> for actions of countable but not necessarily amenable groups.  
 
 
 Consider the set 2^Γ×Γ of binary relations on a countable group Γ, endowed with the product topology.  This topology makes 2^Γ×Γ a compact  metrizable space.
 We will consider the left action Γ2^Γ×Γ  given by
 
 
    x (g · R)y ⇔  (xg) R (yg)  x,y, g ∈ΓR ∈ 2^Γ×Γ.

 
 
 
 
 
 
 
 
 
 Recall that a relation ≺∈ 2^Γ×Γ on Γ is called a partial order if the following requirements hold:
 
 	
 	
  * It is antisymmetric, which means  that  if x ≺ y then  y ⊀x.
 	
  * It is transitive, which means that if x ≺ y and y ≺ z then x ≺ z.
 
 A partial order ≺ is called total  if for any x,y ∈Γ either x ≺ y, y ≺ x or x=y.
 Let (Γ) denote the set of all partial orders on Γ. Denote  the set of all total orders on Γ by (Γ). It is not hard to see that both (Γ) and (Γ) are closed subsets of 2^Γ×Γ. 
 It is easy to see that (Γ) and (Γ) are Γ-invariant subsets.
 
 To a partial order ≺ we associate the past (at the identity):
 
 
 
    Φ_≺ = {γ∈Γ :  1_Γ≺γ}.
 
 A  fixed point for the action Γ(Γ) is called a (deterministic) invariant  order on Γ.
 If ≺ is an  invariant order, then it is straightforward to check that the associated past Φ_≺⊂Γ is a semigroup that does not contain the identity 1_Γ. If  ≺∈(Γ) is a (deterministic) invariant total order, then the  associated past Φ_≺⊂Γ is an algebraic past for Γ, namely it is a semigroup with the property that 
 
    Γ = Φ_≺⊎Φ_≺^-1⊎{1_Γ}.
 
 A group Γ that admits a deterministic invariant total order (or equivalently, admits an algebraic past) is called left-orderable.
 An invariant random order on Γ is a  Γ-invariant Borel probability measure on (Γ).
 An invariant random total order is a Γ-invariant Borel probability measure on (Γ). Equivalently, it is an invariant random order that is supported on the set of total orders.
 Thus, consistently with our notation the space of invariant random orders will be denoted by
 
    _Γ((Γ)),

 and the space of invariant random total orders will be denoted by
 
    _Γ((Γ)).

 
 
 
 
 
 
 
 
 In the sequel we will use the probabilistic convention and write “≺ is an invariant random total order with law ν”
 
  to mean that ≺ is an (Γ)-valued random variable with distribution ν, where 
 
    ν∈_Γ((Γ)).

 In this case for F ∈ L^1(ν) we will use the notation
 
    _≺ F(≺) = ∫ F(≺)dν(≺).

 
 In contrast to deterministic invariant orders, every countable group Γ admits at least one invariant random total order.
 
 Namely, consider the random process (ξ_γ)_γ∈Γ of independent random variables such that each ξ_γ has uniform distribution on [0,1]. Then each realization of this process induces an order on Γ. 
 

 
 We now define what it means for one invariant random total order to extend another:
 Let ν,ν̃∈_Γ((Γ)) be invariant random  orders on Γ. Recall that a joining  of ν and ν̃ is a probability measure    λ∈((Γ)×(Γ)) that is invariant under the Γ-action on the product space and has the property that push-forward of the projection onto the first coordinate coincides with ν and the  push-forward of the projection of λ onto the  second coordinate coincides with ν̃.
 We say that an invariant random order ν̃ extends ν if there exists a joining λ of ν and ν̃
 with the property that
  
 
    λ( { 
     (≺,≺̃) ∈(Γ) ×(Γ) :  
     x ≺ y ⇒ x ≺̃y
     }) =1

  
 	Let  Γ be an amenable group. Then any invariant  random order on Γ 
 	can be extended to an invariant  random total order. 
 
 
 Let ν∈_Γ((Γ)) be an invariant random order.
 Consider the set 
 J(ν) that consists 
 of Borel probability measures  λ∈((Γ) ×(Γ)) whose push-forward via the projection onto the first coordinate is equal to ν and have the property that (<ref>) holds.
 Then J(ν) is a non-empty, compact convex set. Because ν is a Γ-invariant probability measure, the set J(ν) is also  invariant under the natural  
 action of  Γ. By amenability of  Γ, the  action ΓJ(ν) by affine transformations admits a fixed point λ∈ J(ν). It follows that any such fixed point is a joining  of ν with some invariant random total order ν̃ that extends ν. 
 
 
For torsion free locally nilpotent groups, the Rhemtulla-Formanek Theorem <cit.> asserts that any deterministic invariant order  extends to a deterministic invariant total order. Equivalently, for this class of groups any sub-semigroup that does not contain the identity extends to an algebraic past. 
The Rhemtulla-Formanek theorem was  used in <cit.> to prove Theorem <ref> for the class of torsion free locally nilpotent groups. Examples provided in the same paper show that the conclusion of the Rhemtulla-Formanek Theorem fails for more general groups, including  some left-orderable amenable ones.
Lemma <ref> can be viewed as an easy “random substitute” for the Rhemtulla-Formanek theorem that applies to all amenable groups
 
 	Does the statement of Lemma <ref> hold without the amenability assumption on the group?
 

Let  ν∈_Γ((Γ)) be an invariant random order.

Recall that the action ΓX is topologically ν-predictable relative  to π:X → Y if for any f ∈ C(X), and ν-a.e. ≺∈(Γ) the function f is contained in the closed algebra generated by { f∘ g: g ≺ 1_Γ} together with the image of  C(Y) in C(X) under the map  π_*:C(Y) → C(X).


Now 
suppose μ∈_Γ(X) is a Γ-invariant probability measure for the action Γ(X,μ) and 
 ν∈_Γ((Γ)) is an invariant  random order. We say that the measure preserving action Γ(X,μ) is measure-theoretically ν-predictable relative to π:X → Y if for every   countable  Borel partition α with H_μ(α) < ∞ 
we have that for ν-a.e ≺∈(Γ), the partition  α is measurable with respect to the μ-completion of α^Φ_≺∨π^-1((Y)).


Let us introduce the following random generalization for the notion of an algebraic past. An invariant random past on Γ is  a random function Φ̃:Γ→ 2^Γ, or equivalently a 
Borel probability measure  on (2^Γ)^Γ, with the following properties:

	
  (i)  For almost every instance of Φ̃:Γ→ 2^Γ and for all g ∈Γ the condition
	g ∉Φ̃(g) holds.
	
  (ii)
	For almost every instance of Φ̃:Γ→ 2^Γ, for all g,h ∈Γ, if g ∈Φ̃(h) then Φ̃(g) ⊂Φ̃(h). 
		
  (iii) If g  h then either g ∈Φ̃(h) or h ∈Φ̃(g).
	
  (iv) For all g ∈Γ the random subsets
	Φ̃(g) and Φ̃(1_Γ)g have the same distribution.



It follows directly from the definitions that if  ≺ is an invariant random total order, then the random function given by g ↦{h ∈Γ :  h ≺ g} defines 
an invariant random past. If Φ̃ is a random past on Γ with law ν̃∈((2^Γ)^Γ) and F ∈ L^1(ν̃) we use the following probabilistic notation:

 
    _Φ̃ F(Φ̃)  = ∫ F(Φ̃) dν̃(Φ̃).
  
 
 



§ THE  KIEFFER-PINSKER FORMULA


In this section we state and prove a simultaneous but rather straightforward generalization of  Kieffer's entropy formula <cit.>  and of Pinsker's entropy formula for actions of amenable groups <cit.>. 
The earliest and most basic case of this formula for the group Γ = with the usual order goes back to Kolmogorov's very first paper <cit.> on entropy. 



Let Γ(X,μ) be a probability  measure preserving action of a countable amenable group Γ,  
let Φ̃:Γ→ 2^Γ be an invariant random past on Γ with Φ = Φ̃(1_Γ) ⊂Γ. Suppose that
α is a Borel partition with H_μ(α) < +∞ and that

  is a Γ-invariant σ-algebra on X. Then

	the following holds: 
	
    h_Γ(α, X, μ|) = _Φ̃ H_μ(α|α ^Φ∨),







Before the proof we will establish a couple of auxiliary lemmata. We assume that Φ̃:Γ→ 2^Γ, Φ= Φ̃(1_Γ), α and  are as in the statement of Theorem <ref>.



	
	For any ε>0 there is such a finite subset D of Γ so that 
	for any D' ⊃ D
	
	
	
    _Φ̃H_μ(α|α^D' ∩Φ∨) ≤_Φ̃H_μ(α|α^Φ∨) + ε.



	
	Let us consider an arbitrary subset Φ⊂Γ.
	
	Let (D_i)_i=1^∞ be an increasing sequence of finite subsets of Γ such that ⋃ D_i = Γ. 
	A classical argument using  the Martingale convergence theorem and Chung's Lemma (as in  <cit.>) implies that
	
    lim_i →∞ H_μ(α|α^D_i ∩Φ∨) = H_μ(α|α^Φ∨).


	 Using the monotone convergence theorem it follows that for sufficiently big i ∈ 
	
    _Φ̃ H_μ(α|α^D_i ∩Φ∨) ≤_Φ̃ H_μ(α|α^Φ∨) + ε.

	Choose i ∈ that satisfies (<ref>), and let D=D_i. 
	By monotonicity of conditional entropy (as in  <cit.>),  for any superset D' ⊃ D we have
	
    _Φ̃H_μ(α|α^D' ∩Φ∨) ≤_Φ̃H_μ(α|α^D ∩Φ∨).


	
	Thus (<ref>) holds for any   D' ⊃ D. 
	 




	For every finite F ⊂Γ we have
	
    H_μ(α^F |)=∑_g ∈ F_Φ̃
    	H_μ(  α|α^F g^-1∩Φ∨).



	Fix an instance of Φ̃:Γ→ 2^Γ that satisfies properties (i)–(iii) of an invariant random past. 
	Write F as 
	F = { g_1, … , g_|F|} ordered so that g_j ∈Φ̃( g_i) iff i < j. Applying the chain  rule for entropy (as in  <cit.>) we have:
		
    H_μ(α^F|)=
    	∑_i=1^|F|  H_μ( α^g_i|⋁_j < iα^g_j∨)

		This can be rewritten as :
		
    H_μ(α^F|) = ∑_g ∈ F H_μ ( α^g |α^F ∩ (Φ̃(g))∨).

	Taking the expectation over  Φ̃,  using property (iv) of an invariant random  past and linearity of expectation we get (<ref>).




	
	
	
	
	
	
	
	

	Let (F_i)_i=1^∞ be a Følner sequence in Γ.

	By Lemma <ref> we have:
	
    H_μ(α^F_i|) = ∑_g ∈ F_i_Φ̃ H_μ(α|α^F_i g^-1∩Φ∨).

	Choose any  ε>0, and let D ⊂Γ as in 
     Lemma <ref>. Then for any D' ⊃ D we have  that 
    _Φ̃H_μ(α|α^Φ∨)≤_Φ̃H_μ(α|α^D' ∩Φ∨) ≤_Φ̃H_μ(α|α^Φ∨) + ε

	Let ∂_D F_i denote the set of  g ∈ F_i such that D ⊄F_i g^-1.   
	From the definition of a Følner sequence we can derive that 
	
    lim_i →∞|∂_D F_i|/| F_i| = 0.

	For any g ∈ F_i∖∂_D F_i we will have 
	
    _Φ̃H_μ(α|α^Φ∨) ≤_Φ̃H_μ(α|α^F_i g^-1∩Φ∨) ≤_Φ̃H_μ(α|α^Φ∨) + ε

	The latter together with equation (<ref>) implies that
	
    _Φ̃H_μ(α|α^Φ∨)≤lim_i →∞H_μ(α^F_i|)/| F_i|≤_Φ̃H_μ(α|α^Φ∨) + ε.

	This finishes the proof since ε>0 is arbitrary.


	Let Γ be a countable amenable group. Then a probability measure preserving action Γ(X,μ) 
	has zero Kolmogorov-Sinai entropy relative to a factor map π:X → Y if and only if it is measure-theoretically ν-predictable relative to π:X → Y with respect to some (hence any) invariant random  total order.
	
	



	An action  Γ(X,μ) has zero Kolmogorov-Sinai entropy if and only if for any finite measurable partition α we have
	
    h_Γ(α, X, μ|) =0.

	By the Kieffer-Pinkser formula, for any invariant random past this is equivalent to 
	
    _Φ̃ H_μ(α|α ^Φ∨) =0,

	which is equivalent to having H_μ (α|α ^Φ∨) =0 for a.e. realization of Φ̃. 
	This is equivalent to the statement that α is measurable with respect to the μ-completion of α ^Φ∨.


Furthermore, predictability of a relative generator with respect to an invariant  random partial order implies zero relative entropy:

	Let Γ(X,μ) be a measure preserving action of a countable amenable group Γ and let  be a Γ-invariant σ-subalgebra. Suppose that ≺ is an invariant random partial order on   Γ.
	Let α be a finite Shannon entropy partition of X such that α⊂∨α^Φ_≺ for almost every  instance of ≺ and 
	
    α^Γ∨ = (X) μ.

	Then
	h_Γ( X, μ| ) = 0.
	


	Denote the law of the invariant random partial order ≺ by ν∈_Γ((Γ)).
	By Lemma <ref>, we can find   an invariant random total order ν̃∈_Γ((Γ)) that extends ν. Because ν̃ extends ν it follows that α⊂∨α^Φ_≺ for ν̃ -a.e instance of ≺. This implies by Theorem <ref> that
	h_Γ( X, μ| ) = 0.



Seward  <cit.> defined the relative Rokhlin entropy for measure-preserving actions of countable groups.
For an ergodic measure-preserving action Γ(X,μ) of a countable group Γ and Γ-invariant σ-subalgebra, it is given by

    _Γ(X,μ|) = inf_α H_μ(α|),

where α ranges over all generating partitions (countable partitions α that satisfy (<ref>)).
For a free action of an amenable group, Rokhlin entropy coincides with Kolmogorov-Sinai  entropy.
 Seward proved the following far-reaching extension of Sinai's theorem: Any free ergodic measure-preserving action Γ(X,μ) with positive Rokhlin entropy admits a Bernoulli factor, or equivalently it admits a (non-trivial) partition whose iterates under Γ are jointly independent <cit.>. 
 As an immediate  corollary of Seward's  Bernoulli factor theorem we have:


	Let Γ be a countable group, let  ν be an invariant random partial order on Γ. A free ergodic  action  Γ(X,μ) that is measure-theoretically predictable with respect to some  invariant random partial order has zero Rokhlin entropy.


   If a free ergodic action Γ(X,μ) has positive Rokhlin entropy then Seward's Bernoulli factor theorem says that it  admits a non-trivial finite partition α with independent Γ-iterates. Such a partition α is not measurable with respect to the μ-completion of α^Φ_≺, for any order ≺ on Γ. Hence, a free ergodic action  with positive Rokhlin entropy is not measure-theoretically predictable with respect to any invariant random order.


	Is there a direct proof of Proposition <ref> that does not use  Seward's  Bernoulli factor theorem?




§ FROM TOPOLOGICAL PREDICTABILITY TO MEASURE-THEORETICAL PREDICTABILITY VIA  Μ-CONTINUOUS PARTITIONS

In this section we complete the proof of Theorem <ref>.
The steps are essentially identical to  Hochman's proof in  <cit.>, where only the lexicographic past on the group   was considered, without the “relative” version (the image of the factor map π: X → Y was the trivial one-point space).
Just as in Hochman's proof, we will rely on the variational principle for topological entropy.  


	Let ΓX be an action of a countable amenable group by homeomorphisms on a compact metrizable space X. 
	Then the topological entropy h_Γ(X) is given by
	
    h_Γ(X) = sup_μ∈_Γ(X) h_Γ(X,μ).


Note that Kerr and Li proved a more general variational principal for sofic entropy <cit.>.



As in the previous sections,  let X be a compact metric space and let μ be a Borel probability measure on it.
The Rokhlin distance d_μ(α,β) between two partitions α and β of finite Shannon entropy is defined by the formula 

    d_μ(α, β) = H_μ(α|β) + H_μ(β|α).
  
It is well known that for a measure preserving action of an amenable group the Kolmogorov-Sinai entropy is an 1-Lipschitz function with respect to the Rokhlin metric on the space of partition with finite Shannon entropy.


 A partition α of (X,μ) is said to be a μ-continuous partition if there is a continuous function f: X → such that pieces of α are equal to the level sets of f up to μ-null subset. This definition is due to Hochman <cit.>; he proved the following (in the more general setup where X is a normal topological space and  μ is a regular Borel  probability measure): 

	For any Borel probability measure μ∈(X), the  μ-continuous partitions are dense with respect to the Rokhlin metric.




	Let ΓX,ΓY and π: X → Y be as in the statement of Theorem <ref>.
Let 
    = π^-1((Y)) ⊂(X).

	

 Let μ be a Γ-invariant measure on X.
 We note that h_Γ(Y,π(μ)) = h_Γ(,X,μ).
	
	Take any μ-continuous partition α  of (X,μ) with H(α) < ∞. Then topological predictability of the action relative to π:X → Y  implies that  α is measurable with respect to the completion of α^Φ_≺∨ for almost every  instance of the invariant random total order ≺. By Corollary <ref> of Theorem <ref> 
	it follows that
	
    h_Γ(α,X,μ|) =0.

	Since this holds for a set of partitions that is dense with respect to the Rokhlin metric,
	and the function α↦ h_Γ(α,X,μ|) is continuous (in fact 1-Lipschitz with respect to the Rokhlin-metric), it follows that
	
    h_Γ(X,μ|) = 0.

	
	The  Abramov-Rokhlin entropy addition formula for amenable group actions <cit.> asserts that  
	
    h_Γ( X, μ ) = h_Γ( ,X, μ) + h_Γ(X,μ|).

	Thus for any μ∈_Γ(X):
	
    h_Γ(X,μ)= h_Γ(,X,μ)= h_Γ(Y, π(μ)).

	By the variational principle (Theorem <ref>) it follows that 
	
    h_Γ(X) ≤ h_Γ(Y).

	Since the topological entropy for actions of amenable groups is factor-monotone, we have
	
    h_Γ(X) = h_Γ(Y).

	





§ PRIME ACTIONS HAVE ZERO TOPOLOGICAL ENTROPY


An action ΓX is called topologically prime if every factor map is either an isomorphism or it maps onto the trivial action on the one-point space.
More generally, if π:X → Y is a topological factor map between  ΓX and  ΓY, we say that   π:X → Y is a topologically prime extension if ΓX has no intermediate factors.
Equivalently, C(X) has no strict Γ-invariant  C^*-subalgebras that strictly contain π_*(C(Y)).

King constructed and example of a homeomorphism on the Cantor set that is  topologically prime <cit.>, because it has “topological minimal self-joinings” in the sense of del Junco <cit.>. The later property, called “doubly minimal” by Weiss  <cit.>,  means that any pair of points x,y ∈ X that are not in the same orbit have a dense orbit in X × X.  
More generally, any free and ergodic measure-preserving -action with zero entropy  admits a uniquely ergodic  doubly minimal  model  <cit.>. In particular there  is topologically prime  model for any free, ergodic -action with zero entropy <cit.>.







The following result was communicated to us by Benjy Weiss. With his kind permission we reproduce his proof. 


	Suppose that Γ is a countable amenable group that acts on X and Y, and that π:X → Y is a topologically prime extension.
	Then h_Γ(X)=h_Γ(Y).
	

For Γ= and π equal to the trivial factor, a proof of Theorem <ref> appears in <cit.>. See also <cit.>.


Let π: X → Y be a topologically prime extension. Choose any 
Γ-invariant measure μ∈_Γ(X), and let ν∈_γ(Y) be the push-forward of μ via π. We will prove that 

    h_Γ(X,μ)=h_Γ(Y,ν).



We first prove this under the additional assumption that μ satisfies the following property: 

    inf{μ(A) :  A μπ^-1((Y))} =0.






Since μ-continuous partitions are dense with respect to the Rokhlin metric, we can find for any ϵ >0 a μ-continuous partition α  such that 

    0 <H_μ(α|π^-1(Y)) < ϵ.


It follows that

    h_μ(α, X , μ|π^-1(Y))  < ϵ.
  

Since π:X → Y is  a topologically prime extension, for any f ∈ C(X) ∖π_*(C(Y)),
 C(X) is contained in the Γ-invariant C^*-algebra generated by f and π_*(C(Y)).
By the left inequality in (<ref>), α  is not contained in the μ-completion of π^-1((Y)). It follows that 

    (α∨π^-1(Y))^Γ = (X) μ.





By Kolmogorov-Sinai theorem,

    h_Γ(X,μ|π^-1(Y))  < ϵ.

Since ϵ >0 was arbitrary, this shows
that

    h_Γ(X,μ|π^-1(Y))  =0.

By the Abramov-Rokhlin entropy addition formula, this implies (<ref>).

It remains to prove that (<ref>) holds even when μ∈_Γ(X) does not satisfy (<ref>). 
	If (<ref>) does not hold, then there exists ϵ >0 so that every A ∈(X) with μ(A) <ϵ is also measurable with respect to the  μ-completion of π^-1(Y). In this case the factor map π:X → Y  is of the following very degenerate form:
Let μ =p·μ_c + (1-p)·μ_a be the representation of  μ as a convex combination of a  purely continuous measure μ_c ∈_Γ(X)  and purely atomic measure μ_a ∈_Γ(X). Then π:X → Y gives a measure-preserving isomorphism between Γ(X,μ_c) and Γ(Y,π(μ_c)).
In particular,  the factor map π:X → Y  is finite-to-one (actually bounded-to-one) μ-almost everywhere. Because Γ is an infinite amenable group,  finite-to-one extensions do not increase entropy and  (<ref>) follows in this case.





	In the proof above we use the fact that Γ is infinite to conclude that    finite-to-one extensions do not increase entropy.
	If Γ is a finite group, Theorem <ref> fails. To see this, take ΓX to be the action of a finite group Γ on itself by translations, and take ΓY to be an action of Γ on the cosets of a maximal proper subgroup.



We  now consider topologically prime actions of sofic groups. For a measure preserving action Γ(X,μ) of a  sofic group Γ with a sofic approximation Σ, let h_Γ^Σ(X,μ) denote the sofic entropy of the action (for definitions see for instance <cit.>).




The first-named author and Brandon Seward  <cit.> and independently Ben Hayes <cit.>  proved  the following Abramov-Rokhlin-type inequality:




	Let Γ be a sofic group with sofic approximation Σ, and let π:X → Y be a factor map between the measure preserving actions  Γ(X,μ) and Γ(Y,μ). If α is a measurable partition of X such that
	
    (α∨π^-1((Y)))^Γ⊃(X) μ,

	then
	
    h_Γ^Σ(X,μ) ≤ h_Γ^Σ(Y,ν) + H_μ(α|π^-1((Y)).


The above  is a simplified and slightly less general form of the corresponding statements from <cit.>.

The following proposition on sofic entropy of finite-to-one extension follows from the main theorem of Hayes' work <cit.> relating sofic entropy and spectral properties of actions. That work establishes much more general results. In particular, the statement below will hold with compact extensions instead of finite-to-one.  See the discussion following Proposition 5.7 in <cit.>. The case needed in our exposition allows for a short combinatorial proof, which we include for the sake of self-containment.


	Let Γ be a countably infinite sofic group with sofic approximation Σ, and let π:X → Y be a finite-to-one factor map between the measure preserving actions  Γ(X,μ) and Γ(Y,ν). Then 
    
    h_Γ^Σ(X,μ)≤ h_Γ^Σ(Y,ν).



	
	Because π:X → Y is finite-to-one, by removing  null sets we can assume that  |π^-1({y})| is a positive integer for every y ∈ Y.
	For every n ∈ let 
    Y_n = {y ∈ Y: π^-1({y}) =n} X_n = π^-1(Y_n).

	
Let us first assume in addition that Γ(Y,ν) is aperiodic.
By the Abramov-Rokhlin sub-addition formula for sofic entropy (Proposition <ref>) it is enough to show that for every ϵ >0 we can find a measurable partition α of X that satisfies (<ref>)




and H_μ(α) < ϵ.
Because Γ(Y,ν) is aperiodic, so is  Γ(X,ν). So (X,μ) is a standard probability space with no atoms.
Let ϕ:X → [0,1] be a Borel bijection. 

Because Γ(Y,ν) is aperiodic  whenever ν(Y_n) >0 we can find arbitrary small ϵ_n >0 and  a Borel measurable set A_n ⊂ Y_n so that ⋃_g ∈Γg(A_n)= Y_n and  ν(A_n)=ϵ_n (for instance by considering the ergodic decomposition of Γ(Y,ν)). 
Because π:X → Y is finite-to-one  we can define ψ:X → by

    ψ(x)= #{x' ∈π^-1({π(x)}): ϕ(x')  ≤ϕ(x) }.

Then for every y ∈ Y_n,  ψ induces a bijection between  π^-1({y}) and {1,…,n}.
We denote the  inverse  by 
    ψ_y^-1:{1,…,n}→π^-1({y}).

Also we have  a Borel bijection Ψ:X →⋃_i=1^∞(Y_n ×{1,…,n})
given by

    Ψ(x)= (π(x),ψ(x)).

For y ∈ Y_n and g ∈Γ define a permutation Π_g,y of {1,…,n}
by

    Π_g,y(i) = ψ(g(ψ_y^-1(i)))  i ∈{1,…,n}.

Then the map (g,y) ↦Π_g,y is a Borel map from Γ× Y_n to the set of permutations on {1,…,n}.




For every n ∈ and i ∈{1,…,n} let 

    B_n,i= π^-1(A_n) ∩ψ^-1({i}).

Consider the partition α of X given by

    α ={ B_n,i:  n ∈ 1 ≤ i ≤ n }∪{X ∖⋃_n ∈π^-1(A_n)}.

It follows that  

    H_μ(α) 
    
    < ∑_n=1^∞ϵ_n(log(ϵ_n^-1)+log(n) )+log(1-∑_n=1^∞ϵ_n),

So if (ϵ_n)_n=1^∞ are sufficiently small H_μ(α) < ϵ.
In order to check that α is a relative generator in the sense that it satisfies (<ref>), it suffices to check that 
ψ:X → is α^Γ∨π^-1((Y))-measurable. Indeed, because Y_n = ⋃_g ∈Γg (A_n) for any x ∈ X_n there exists g ∈Γ and i  ∈{1,…,n} so that g(x) ∈ B_n,i. For such x we have
 ψ(g(x))=i so ψ^-1_g(π(x))(i)=g(x) so 

    ψ(x)= ψ(g^-1(g(x)))=ψ(g^-1(ψ^-1_g(π(x))(i)))=Π_g^-1,π(g · x)(i).

This implies the desirable measurability.






Now let us remove the additional assumption that Γ(Y,ν) is aperiodic.
Let Γ(Z,η) be a Bernoulli action.
Then π:X → Y naturally induces a finite-to-one factor map from Γ(X × Z,μ×η) to Γ(Y × Z,ν×η). Since Γ(Y × Z,ν×η) is aperiodic we can use the first part to conclude that 

    h_Γ^Σ(X × Z, μ×η) ≤ h_Γ^Σ(Y × Z, μ×η).

By  <cit.>, because (Z,η) is a  Bernoulli action

    h_Γ^Σ(X × Z, μ×η)= h_Γ^Σ(X,μ)+h_Γ^Σ(Z,η),

and

    h_Γ^Σ(Y × Z, μ×η)= h_Γ^Σ(Y,μ)+h_Γ^Σ(Z,η).

This proves (<ref>) without assuming that  Γ(Y,ν) is aperiodic.




	Suppose Γ is a sofic group with sofic approximation Σ that acts on X and Y and that  π:X → Y is a topologically prime extension.
	 Then 
	
    h_Γ^Σ(X)≤ h_Γ^Σ(Y).

	In particular, topologically prime actions of sofic groups have non-positive sofic entropy.
	


We obtain the proof essentially by repeating the proof of Theorem <ref>, with the following modifications:
Instead of applying the Abramov-Rokhlin entropy addition formula we apply the  Abramov-Rokhlin sub-addition formula for sofic entropy (Proposition <ref>). To deal with the case where (<ref>) fails, we apply Proposition <ref> above.








We note that in general we cannot  conclude an equality instead of  the equality (<ref>) under the assumptions of Theorem <ref>. 

From the definition of sofic entropy, if XΓ admits no invariant probability measures then h_Γ^Σ(X) = - ∞.  
The following is an example of a topologically prime action of the free group on two generators that admits no invariant probability measure:


	Let T_1:X → X be a topologically prime homeomorphism that is uniquely ergodic (for instance Kings's example <cit.> on the cantor set X), and let T_2:X → X be a homeomorphism that does not preserve the unique T_1-invariant measure. Consider the action on X of the free group generated by T_1 and T_2. This is a topologically prime action because it has T_1 as a subaction. It  admits no invariant probability measure, because the unique T_1-invariant measure is not T_2-invariant.



	Although finite-to-one  extensions can never increase sofic entropy  they can certainly decrease it, as in  the well known Ornstein-Weiss example of a two-to-one factor map from the Bernoulli 2-shift to the Bernoulli 4-shift over the free group. 
	


	Using the Abramov-Rokhlin sofic entropy sub-addition formula and similar arguments  as in the proof of Proposition <ref> it is possible  to prove the following “atomless” refinement of the Abramov-Rokhlin entropy sub-addition formula as follows: If μ= p μ_c + (1-p)μ_a is the representation of μ as a convex combination of a continuous measure μ_c and a purely atomic measure μ_a and
    α is a measurable partition of X such that
	
    (α∨π^-1((Y)))^Γ⊃(X) μ_c,

	then
	
    h_Γ^Σ(X,μ) ≤ h_Γ^Σ(Y,ν) + H_μ(α|π^-1((Y)).

	



abbrv

