Topological Pressure for the Completely Irregular Set of Birkhoff Averages

In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic average from one (or finite) observable function to any number of observable functions from the dimensional perspective. For any topological dynamical system with $g-$almost product property and uniform separation property, we show that any {\it jointly-irregular set}(i.e., the intersection of a series of $\phi-$irregular sets over several continuous functions) either is empty or carries full topological pressure. In particular, if further the system is not uniquely ergodic, then the {\it completely-irregular set}(i.e., intersection of all possible {\it nonempty $\phi-$irregular} sets) is nonempty(even forms a dense $G_\delta$ set) and carries full topological pressure. Moreover, {\it irregular-mix-regular sets} (i.e., intersection of some $ \phi-$irregular sets and $ \varphi-$regular sets) are discussed. Similarly, the above results are suitable for the case of BS-dimension. As consequences, these results are suitable for any system such as shifts of finite type or uniformly hyperbolic diffeomorphisms, time-1 map of uniformly hyperbolic flows, repellers, $\beta-$shifts etc..


Introduction
Let f be a continuous map of a compact metric space X and let φ : X → R be a continuous observable function. A point x ∈ X is called to be φ-regular, if the limit exists. Otherwise, x is called φ-irregular(or, we say the orbit of x has historic behavior). Define the φ-regular set to be the set of all φ-regular points, that is, and define the φ-irregular set to be the set of all φ-irregular points, that is, By Birkhoff's ergodic theorem, the φ−irregular set is always of zero measure for any invariant measure. That is, µ(I(φ, f )) = 0 for all invariant measure µ. Let where C 0 (X) denotes the space of all continuous functions on X. I(f ) is called irregular set, composed of all φ-irregular points for all continuous functions. Its complementary set X \ I(f ) is called regular set, denoted by R(f ). Every point in R(f ) is φ−regular for all continuous functions. That is, R(φ, f ).
Pesin and Pitskel [24] are the first to notice the phenomenon of the irregular set carrying full topological entropy in the case of the full shift on two symbols from the dimensional perspective. Barreira, Schmeling, etc. studied the irregular set in the setting of shifts of finite type and beyond, see [6,4,34,11,36,35,29,37,23] etc. Ruelle uses the terminology in [30] 'historic behavior' to describe irregular point and in contrast to dimensional perspective, Takens asks in [33] for which smooth dynamical systems the points with historic behavior has positive Lebesgue measure. Moreover, many researchers studied irregular set from topological or geometric viewpoint that irregular set forms dense G δ set, see [19,18,2,16,22,3] etc. So many people paid attention to the irregular set and the study of irregular set is an increasingly hot topic.
In current work, we still take the dimensional viewpoint. Firstly we recall a recent dimensional result from [35,36] that which is inspired from [26] by Pfister and Sullivan and [34] by Takens and Verbitskiy) that Theorem 1.1. Let f be a continuous map of a compact metric space X with (almost) specification. Then for any continuous function φ : X → R, the φ−irregular set I(φ, f ) either is empty or carries full topological entropy(or pressure). Theorem 1.1 studied the φ−irregular set under one observable function. A natural and general question is, how about the intersection of φ−irregular sets under several observable functions(called jointly-irregular set)? Let us recall a result of [6](Theorem 2.1) that Theorem 1.2. For systems such as topological mixing subshifts of finite type (repellers and horseshoes), the jointly-irregular set under finite observable Hölder functions φ 1 , φ 2 , · · · , φ k (k ≥ 1) either is empty or carries full topological entropy.
The special case of Theorem 1.1 that I(f ) = φ∈C 0 (X) I(φ, f ) carries full topological entropy, was firstly proved in [11] for systems with specification property. Theorem 1.1 is a refined result and Theorem 1.2 is much more refined. In present paper, we are going to give a substantial generalization of Theorem 1.2 in three directions: (1) the system is more general; (2) the observable functions are not necessarily Hölder continuous, (3) the number of observable functions can be infinite. In particular, Theorem 1.1 can be generalized from one observable function to any number of observable functions. More precisely, we state the question as follows: Question 1.3. Let f be a continuous map of a compact metric space X with (almost) specification. For any subset D ⊆ C 0 (X), whether the jointly-irregular set φ∈D I(φ, f ) either is empty or has full topological entropy(or pressure)?
We conjecture that it is true. To answer Question 1.3, in general the fundamental idea is to construct lots of irregular points such that you can use 'so many points' to prove full topological entropy (or pressure). However, in the new setting, we need to construct lots of irregular points which are irregular for a series of functions simultaneously. So the main problem is how to deal with infinite functions. Inspired from [26] by Pfister and Sullivan, we can use variational principle to solve the problem but another condition, called uniform separation property, is required. So we can give a positive answer in present paper under an additional assumption, uniform separation property. Fortunately, it can be applicable to lots of classical dynamical systems, including all uniformly hyperbolic diffeomorphisms, time-1 map of uniformly hyperbolic flows, repellers, shifts of finite type and β−shifts etc.. Along this paper we will state and prove several theorems that give partial positive answers to Question 1.3. The results are classified in two groups: Theorem 1.2 for systems from symbolic dynamics to more general case and for observable functions from Hölder continuous to just continuous.
• The general case and the particular "perfect" case: Remark that if D contains a constant function or a function ϕ cohomologous to a constant(i.e., there is a constant c and a continuous function h such that ϕ = c + h − h • f ), then the jointlyirregular set So it is meaningful only for considering the functions with I(φ, f ) = ∅. A natural and "perfect"jointly-irregular set is the intersection of all nonempty φ−irregular sets, called completely-irregular set. This set is 'minimal' in the sense of possible non-emptiness so that if this case is true, then so does the Question 1.3. For systems with g−almost product property and uniform separation property, we gives a positive answer(see Theorem 1.4 and Theorem 1.8).
Remark that the case of finite observable functions does not imply the general case, because one can not find finite observable functions such that the completely-irregular set is a jointly-irregular set under the observation of such functions, see Proposition 3.3. Moreover, we point out that it is still unknown wether the completely-irregular set can be written as a jointly-irregular set under the observation of countable functions. 3 ) Let f be a continuous map of a compact metric space X with g−almost product property and uniform separation property. Then for any subset D ⊆ C 0 (X), the the jointly-irregular set φ∈D I(φ, f ) either is empty or carries full topological pressure, that is,

Jointly-irregular set of any number of observable functions
for any ϕ ∈ C 0 (X). Remark 1.5. If f is uniquely ergodic, there is an ergodic measure µ such that for any x ∈ X, 1 n n−1 i=0 δ f i (x) converges to µ in weak * topology, where δ y denotes the Dirac measure supported on the point y ∈ X. Then for any x ∈ X and any φ ∈ C 0 (X), That is, for any φ ∈ C 0 (X), I(φ, f ) = ∅. Thus in this case the jointly-irregular set φ∈D I(φ, f ) is always empty.
Notice that only the functions with I(φ, f ) = ∅ is meaningful. So we only need to consider the case of D which is composed of several functions with I(φ, f ) = ∅. For convenience, Definition 1.6. (Truly-observable) We call a continuous function φ ∈ C 0 (X) to be truly-observable, if That is, there is at least one orbit of some x has historic behavior under the observation of φ. Otherwise, φ is called trivially-observable.
For example, any constant function is naturally trivially-observable. Roughly speaking, every truly-observable function can open an 'eye' to observe irregular points. Let For example, the south-north map and uniquely ergodic systems are all such kind. Let us introduce an important kind of jointly-irregular set, called completely-irregular set(or called essentially-irregular). More precisely, Definition 1.7. (Completely-irregular set) define completely-irregular set as In other words, the completely-irregular set CI(f ) is the intersection of all possible nonempty φ−irregular sets.
Remark that it is the 'minimal' jointly-irregular set in the sense of nonempty possibility, since if one add one more function the jointly-irregular set must be empty. The element of completely-irregular set is called completely-irregular point. That is, a point x is completely-irregular, if x is φ−irregular for any φ ∈Ĉ 0 f (X). Roughly speaking, every completely-irregular point is persistent under all trulyobservable functions. In other words, every completely-irregular point can be observed by the 'eyes' of all truly-observable functions. Thus, if we think the regular set R(f ) as the 'best' dynamical set, then CI(f ) is the 'worst' dynamical set in opposition. Remark that for any φ ∈Ĉ 0 f (X) , Inspired by the analysis of Remark 1.5, we only need to consider the system which is not uniquely ergodic. Let D ⊆ C 0 (X) be a subset of continuous functions. Remark that if φ∈D I(φ, f ) = ∅, then φ∈D I(φ, f ) ⊇ CI(f ).
Then Theorem 1.4 can be deduced from following theorem. Theorem 1.8. ( Full Pressure of Completely-irregular set) Let f be a continuous map of a compact metric space X with g−almost product property and uniform separation property. Assume that f is not uniquely ergodic. Then the completely-irregular set CI(f ) is nonempty and carries full topological pressure, that is, For non-uniquely ergodic systems with g−almost product property(or almost specification) CI(f ) is nonempty from Theorem 2.17 below. However, if f has positive topological entropy, then the conclusion of full topological entropy(ϕ = 0) of Theorem 1.8 implies the non-emptiness of CI(f ) . More precisely, for ϕ = 0, CI(f ) has positive full entropy so that it is not empty.
We will prove Theorem 1.4 and Theorem 1.8 in Section 3.2.

Jointly-irregular set of finite observable functions
Notice that Theorem 1.8 requires the condition of uniform separation. Here we drop the uniform separation property and try to prove a theorem as a weaker version of Question 1.3. That is, we consider the jointly-irregular set which is the intersection of finite φ−irregular sets. Theorem 1.9. (Partial Positive Answer to Question 1. 3) Let f be a continuous map of a compact metric space X with (almost) specification. Then for any finite functions φ 1 , · · · , φ k ∈ C 0 (X)(k ≥ 1), the jointly-irregular set ∩ k j=1 I(φ j , f ) either is empty or carries full topological pressure, that is, If f has uniform separation property, then this theorem can be deduced immediately from Theorem 1.4. But here it is enough to assume f satisfying (almost) specification. Remark that Theorem 1.1 is a particular case of Theorem 1.9. We will prove Theorem 1.9 in Section 3.4.

BS-dimension
By the definitions of topological pressure and BS-dimension(see Section 2.1 below for the definitons), it is not difficult to see that for any set Z ⊆ X, the BS-dimension of Z is a unique foot of Bowen's equation P (Z, −sϕ) = 0, i.e., s = BS(Z, ϕ).
(1.1) So all results in present paper for topological pressure can be also realized for the case of BS-dimension. For example, the conclusion of Theorem 1.8 can be stated for any strictly positive function ϕ ∈ C 0 (X).

Applications
The above consequence can be applicable to lots of dynamical systems(that is, Question 1.3 is true for these systems). For example, Theorem 1.10. For any one of following systems, the completely-irregular set CI(f ) is not empty and carries full topological pressure(in particular, topological entropy) and BS-dimension: (B). f : X → X is a subsystem of an Axiom A system f : M → M over a compact Riemannian manifold M where X is a hyperbolic elementary set. (B'). f : X → X is a transitive Anosov diffeomorphism of a compact Riemannian manifold X.
(C). f : X → X is the time-t map(t = 0) of a transitive Anosov flow of a compact Riemannian manifold X(in this case, f is partially hyperbolic).
(D). f : X → X is a subsystem of a C 1 map f : M → M over a compact Riemannian manifold M where X is a topological mixing and expanding invariant set(called repeller).
Remark that this result for the case of shifts of finite type can be as a generalization of [6](Theorem 2.1) in three directions: (1) the observable function are not necessarily Hölder continuous, (2) the number of observable functions can be infinite and (3) topological entropy are replaced by more general concept, topological pressure. Recall a result that for C 1+δ conformal repellers, it was proved in [15] that the jointly-irregular set of finite observable functions is either empty or carries full Hausdorff dimension. In this case if ϕ = log df , then for every Z ⊆ X. So Theorem 1.10 (D) implies that Corollary 1.11. Let f : X → X be a subsystem of a conformal C 1+δ map f : M → M over a compact Riemannian manifold M where X is a topological mixing and expanding invariant set(called conformal repeller). Then for any subset D ⊆ C 0 (X), the the jointlyirregular set φ∈D I(φ, f ) either is empty or carries full Hausdorff dimension, that is, In other words, this corollary generalizes the result of [15] from observation of finite functions to any number of functions.
It is known that any system f in Theorem 1.10 is not uniquely ergodic, has positive entropy and satisfies specification(topological mixing + shadowing property ⇒ specification). From [26] we know that specification implies g−almost product property(Proposition 2.1 in [26]). Note that the systems of (A)-(B') and (D) are all expansive. The system f of (C) is partially hyperbolic with one dimensional central bundle and thus f is far from tangency so that f is entropy-expansive from [20](or see [13,27]). Recall that from [21] entropy-expansive implies asymptotically h−expansive and from [26](Theorem 3.1) any expansive or asymptotically h−expansive system satisfies uniform separation property. Thus, Theorem 1.10 can be deduced from Theorem 1.8.
Let us recall the definition of β−shift(β > 1) (Σ β , σ β ) in [38](Chapter 7.3). We only need to consider that β is not an integer. Consider the expansion of 1 in powers of denotes the integral part of t ∈ R. Let k = [β] + 1. Then 0 ≤ a n ≤ k − 1 for all n so we can consider One can obtain the two-sided β−shift by lettinĝ The topological entropy of β−shift(β > 1) is log β. Remark that by Variational Principle, there is an ergodic measure with positive entropy. Note that the Dirac measure supported on the fixed point x = {0} ∞ 1 ∈ Σ β has zero entropy. So every β−shift is not uniquely ergodic.
It is known that every β−shift(β > 1) (Σ β , σ β ) is expansive(as a subshift of finite type) and satisfies g−almost product property from [26](see the Example on P.934). So the completely-irregular set of every β−shift has full topological entropy(= log β). Moreover, from [36] has full topological entropy, topological pressure, BS-dimension and full Hausdorff dimension.

Some definitions
Firstly we recall the definition of (almost) specification, see [12,31,8,9,26,35,36]. Let f be a continuous map of a compact metric space X.
Definition 2.1. We say that the dynamical system f satisfies specification property, if the following holds: for any ǫ > 0 there exists an integer M(ǫ) such that for any k ≥ 2, any k points x 1 , · · · , x k , any integers The original definition of specification, due to Bowen, was stronger.
Definition 2.2. We say that the dynamical system f satisfies Bowen's specification property, if under the assumptions of Definition 2.1 and for any integer We recall a result that systems with specification naturally have positive entropy. Thus if the system in Theorem 1.8 has specification property, positive entropy is a natural condition. In particular, if the Bowen's specification holds, then the system is not uniquely ergodic. Proposition 2.3. Let f be a continuous map of a compact metric space X with specification property. Assume card X > 1. Then f has positive entropy. In particular, if the specification property is Bowen's specification, then f is not uniquely ergodic.
Proof. Positive entropy is from Proposition 21.6 of [12]. If Bowen's specification holds, from Proposition 21.3 of [12] we know that periodic points are dense in X. Notice that positive entropy implies that X is uncountable. So there are two periodic points with different orbits. Then the measures supported on the two orbits are two different periodic measures(which are all invariant and ergodic). Thus f is not uniquely ergodic.
Recall that almost specification introduced in [36] is slightly different from g−almost product property in [26]( Almost specification is slightly weaker). And their main ideas are same: one requires only partial shadowing of the specified orbit segments, contrary to specification property. Therefore, in present paper we treat almost specification same as g−almost product property and we only introduce the definition of g−almost product property as follows. People who want to know the detailed difference, see [36,26]. A striking and typical example of g−almost product property (and almost specification) is that it applies to every β−shift [36,26]. In sharp contrast, the set of β for which the β−shift has specification property has zero Lebesgue measure [10,32]. The function g is called blowup function. Let x ∈ X and ε > 0. The g−blowup of B n (x, ε) is the closed set Definition 2.5. We say that the dynamical system f satisfies g−almost product property with blowup function g, if there is a nonincreasing function m : R + → N, such that for any k ≥ 2, any k points x 1 , · · · , x k ∈ X, any positive ε 1 , · · · , ε k and any integers where M 0 := 0, M i := n 1 + · · · + n i , i = 1, 2, · · · , k − 1.
Now let us to recall the definition of topological entropy.
Now we recall the definition of uniform separation property. Let M x (f ) be the set of all limits of 1 n For δ > 0 and ε > 0, two points x and y are (δ, n, ε)−separated if A subset E is (δ, n, ε)−separated if any pair of different points of E are (δ, n, ε)−separated. Let ξ = {V i | i = 1, 2, · · · , k}, be a finite partition of measurable sets of X. The entropy of ν ∈ M(X) with respect to ξ is We write f ∨n ξ := ∨ k∈Λ f −k ξ. The entropy of ν ∈ M f (X) with respect to ξ is h(f, ν, ξ) := lim Definition 2.7. We say that the dynamical system f satisfies uniform separation property, if following holds. For any η > 0, there exist δ * > 0, ǫ * > 0 such that for µ ergodic and any neighborhood F ⊆ M(X) of µ, there exists n * F,µ,η , such that for n ≥ n * F,µ,η , Now we recall the definition of topological pressure and entropy. Let E ⊆ X, ϕ ∈ C 0 (X) and F n (E, ǫ) be the collection of all finite or countable covers of E by sets of the form B m (x, ǫ) with m ≥ n. We set C(E; t, ϕ, n, ǫ, f ) := inf{

2.2
Variational principle and some useful lemmas Firstly we recall a result from [26,28]( [26] being for topological entropy and [28] being for topological pressure). We say that f : X → X is saturated, if for any ϕ ∈ C 0 (X) and any compact connected nonempty set K ⊆ M f (X),

Moreover, by Ergodic Decomposition theorem,
inf Lemma 2.9 is a general and direct result from the definition of φ−irregular set and can be stated more precise as one direction of following lemma. Lemma 2.10. Let f be a continuous map of a compact metric space X. Let φ ∈ C 0 (X) and x ∈ X. Then Proof. On one hand, fix φ ∈Ĉ 0 f (X) and x ∈ I(φ, f ). By definition there are two sequences of n j , m j ↑ +∞ such that the following limits exist and lim j→∞ 1 n j By weak * topology one can take two convergence subsequences(if necessary) of and then the two limits of µ 1 and µ 2 are in M x (f ) and satisfy that On the other hand, Let φ ∈ C 0 (X) and x ∈ X satisfy Then we can take two convergence subsequences of such the limits are µ 1 and µ 2 . So Hence, x ∈ I(φ, f ) and thus φ ∈Ĉ 0 f (X). Remark that for systems with (almost) specification, the inverse case of Lemma 2.9 is also true from [36,35].
Lemma 2.11. Let f be a continuous map of a compact metric space X with (almost) specification. Let φ ∈ C 0 (X). Then Proof For the case of '⇒', see the paragraph behind of Lemma 2.1 in [36], as a corollary of Lemma 2.1 and Theorem 4.1 there, P. 5397(see Lemma 1.6 of [35] for the case of specification).
For the case of '⇐', it is our above Lemma 2.9.
Recall another result that g−almost product property implies entropy-density(Theorem 2.1 in [25]). Lemma 2.12. Let f be a continuous map of a compact metric space X with g−almost product property. Then f has entropy-dense property, that is, for any ν ∈ M f (X), any neighborhood G ⊆ M(X) of µ and any h * < h ν (T ), there exists an ergodic measure The following lemma is from [26] (Proposition 3.3).
is upper continuous.

Cardinality of truly-observable functions
If one only has finite truly-observable functions, then Question 1.3 is possibly easier to answer. However, we show that the set of truly-observable functions is uncountable for any non-uniquely ergodic system with (almost) specification.
Proposition 2.14. Let f be a continuous map of a compact metric space X with (almost) specification. If it is not uniquely ergodic, then the set of truly-observable functions, , is open and dense in C 0 (X).
By continuity of sup norm, we can take an open neighborhood of φ, denoted by U(φ), such that for any ϕ ∈ U(φ) ϕdµ 1 < ϕdµ 2 .
Now we start to prove Proposition 2.14.
Proof of Proposition 2.14 By Lemma 2.15, we only need to prove thatĈ 0 f (X) = ∅. By assumption, there are two different invariant measures µ 1 , µ 2 . By weak * topology, there is a continuous function φ such that By Lemma 2.11, I(φ, f ) = ∅.

Some topological properties of completely-irregular set
In a Baire space, a set is residual if it contains a countable intersection of dense open sets. Some results showed that certain irregular sets can also be large from the topological point of view. For example, Albeverio, Pratsiovytyi and Torbin [2], Hyde et al [16] and Olsen [22] proved that some kinds of irregular sets associated with integer expansion are residual. Baek and Olsen [3] discussed the set of extremely non-normal points of selfsimilar set from the topological point of view. Li and Wu [18] proved that the set of divergence points of self-similar measure with the open set condition is either residual or empty, and they also proved in [19] that Theorem 2.16. Let f be a continuous map of a compact metric space X with specification. Then for any continuous function φ : X → R, the φ−irregular set I(φ, f ) either is empty or residual in X.
Now we restart to study Question 1.3 in geometric or topological perspective and find that completely-irregular set is very still "large". It can be as a generalization of Theorem 2.16.
Theorem 2.17. Let f be a continuous map of a compact metric space X with (almost) specification. Assume that f is not uniquely ergodic. Then the completely-irregular set CI(f ) is residual in X ( τ ∈M f (X) Support(ν)).

Proof.
Recall thatĈ 0 f (X) = ∅ is from Lemma 2.15. Now we start to prove the residual property. Firstly we give a proof for any system with Bowen's specification. Recall from [12] that the set of points with maximal oscillation G max := {x ∈ X| M x (f ) = M f (X)} is residual in X(Proposition 21.18 and Proposition 21.14 in [12]). One only needs to show that More precisely, for given φ ∈Ĉ 0 f (X), by Lemma 2.10 there are two invariant measures µ 1 , µ 2 such that For any z ∈ G max , by definition there are two subsequences of converging to µ 1 , µ 2 . Then by weak * topology φdµ 1 = φdµ 2 implies that the limit does not exist. That is, z ∈ I φ . This completes the proof.
For the case of specification, the system is topologically mixing so that one can adapt the proof of [12](Proposition 21.18 and Proposition 21.14 in [12]) to get that G max is residual in X. Notice that is a basic fact without any assumption. Therefore, Theorem 2.17 is true for systems with specification. For convenience of readers, we state a rough idea to prove that G max is residual in X. From [12] it is a general fact that G max is a G δ set. Existence of a point being in G max is to construct a point as follows. By entropy-density of ergodic measures(Lemma 2.12), ergodic measures are dense in the space of invariant measures. Since M f (X) is a compact metric space, one can take a countable subset F , composed by ergodic measures, such that F is dense in M f (X). For any ergodic measure ν ∈ F , choose a generic point x ν which represents the "information" of ν. That is, E n (x ν ) converges to µ in weak * topology. Then by specification and by induction there is a point x shadowing the countable orbits of generic points more and more close(replacing the roles of periodic measures or orbits in [12] by ergodic measures or generic points). Then every ν ∈ F can be as a limit point of E n (x). By density of F , and thus x is the needed point. Density is from topologically mixing, because the needed shadowing point x can be chosen to shadow beginning from any given point in X.
For the case of almost specification, it is not sure that G max is residual in X. But similar as the case of specification, it is possible to adapt the proof of [12](Proposition 21.18 and Proposition 21.14 in [12]) to show that G max is residual in The proof of existence of a point x in G max is similar as the case of specification. Density of G max in ∆ is from the definition of G max , because from weak * topology the support of any invariant measure is contained in the closure of the orbit of x. Here we omit the details, see [12] for possible modification to prove. 19. Let f be a continuous map of a compact metric space X with (almost) specification. Assume that f is not uniquely ergodic. Then for any x ∈ CI(f ), where ω f (x) denotes the ω−limt set of x. In other words, µ(ω f (x)) = 1 holds for any x ∈ CI(f ) and any invariant measure µ ∈ M f (X). In particular, if there is an invariant measure with full support(for example, the specification is Bowen's specification), then for any x ∈ CI(f ), ω f (x) = X.
Proof. Assume by contradiction that there is a point x ∈ CI(f ), This implies that there exists an invariant measure µ and a point y ∈ Support(ν) \ ω f (x).
If the specification is Bowen's specification, from [12] (Proposition 21.12) a dense G δ subset of invariant measures has support X. So and thus ω f (x) = X.

Irregular-mix-regular set
Recall that for any φ ∈Ĉ 0 f (X), CI(f ) ⊆ I(φ, f ) ⊆ I(f ). Since the topological entropy of CI(f ) and I(φ) are studied, a natural interest is the complementary set It is the gap between completely-irregular and irregular. Remark that Inspired by this analysis we introduce a concept called irregular-mix-regular. A set A ⊆ X is irregular-mix-regular, if A is formed from sets such as I(φ, f ) and R(ψ, f ) throughout the operation of intersection. Firstly we discuss I(φ, f ) ∩ R(ψ, f ) under two observable functions.
Theorem 3.1. Let f be a continuous map of a compact metric space X with g−almost product property and uniform separation property. Then for any two continuous functions φ, ψ ∈ C 0 (X), the irregular-mix-regular I(φ, f ) ∩ R(ψ, f ) either is empty or has full topological entropy.
In particular, for any φ ∈ C 0 (X), I(φ, f ) \ CI(f ) either is empty or has full topological pressure.
If ψ is a constant function or a function cohomologous to a constant, obviously R(ψ, f ) = X and thus I(φ, f ) ∩ R(ψ, f ) = I(φ, f ). Now we give a simple example such that Example 3.2. Let f be a continuous map of a compact metric space X with Bowen's specification. Fix three different periodic orbits Orb(p 1 ), Orb(p 2 ), Orb(p 3 ). Take φ to be a continuous function such that and take ψ to be a continuous function such that ψ| Orb(p 3 ) = ψ| Orb(p 2 ) = 0, ψ| Orb(p 1 ) = 1.
Let µ 1 , µ 2 , µ 3 denote the periodic measure supported on Orb(p 1 ), Orb(p 2 ), Orb(p 3 ) respectively. Take K = {τ µ 2 + (1 − τ )µ 3 | τ ∈ [0, 1]}. Then from [12] specification implies that G K is nonempty and dense in X, where G K = {x ∈ X| M x (f ) = K}(Proposition 21.14 in [12]). It is easy to check that In general, we show that for any system with g−almost product property, every truly-observable continuous function φ satisfies I(φ, f ) \ CI(f ) = ∅. Proposition 3.3. Let f be a continuous map of a compact metric space X with g−almost product property(or almost specification). Assume that f is not uniquely ergodic. Then for any finite functions φ 1 , · · · , φ k ∈Ĉ 0 f (X)(k ≥ 1), one has If further uniform separation property holds, the sets CI(f ) and ∩ k j=1 I(φ j , f ) \ CI(f ) both have full topological entropy (deduced from Theorem 1.8 and 3.7). Remark 3.4. This proposition implies CI(f ) is not a jointly-irregular set of finite observable functions.
Proof. By the observation of (3.3) we only need to find some ψ ∈Ĉ 0 f (X) such that Before that we need the first part of Proposition 2.3 of [25].
Lemma 3.5. Let f be a continuous map of a compact metric space X with g−almost product property and µ be an ergodic measure. Then for any neighborhood G ⊆ M(X) of µ, there exists a closed f -invariant subset Y ⊆ X and an integer N G > 0 such that for any n ≥ N G and y ∈ Y, E n (y) ∈ G.
Recall µ 1 , µ 2 to be the two ergodic measures in the proof of Theorem 1.9 and for any 1 ≤ j ≤ k, By entropy density, one can choose another ergodic measure µ 3 close to 1 2 (µ 1 + µ 2 ) enough in weak * topology such that for any 1 ≤ j ≤ k, Take three closed neighborhoods G 1 , G 2 , G 3 of µ 1 , µ 2 , µ 3 such that G 1 , G 2 , G 3 are pairwise disjoint and for any measures ν i ∈ G i (i = 1, 2, 3), Then by Lemma 3.5 one can take three closed f -invariant subsets Y 1 , Y 2 , Y 3 ⊆ X and a common integer N such that for any n ≥ N and y ∈ Y i , Remark that Y 1 , Y 2 , Y 3 are pairwise disjoint so that there exists a continuous function ψ : Take ν i (i = 1, 2, 3) to be three ergodic measures supported on Y i . Then by Birkhorff ergodic theorem, each ν i is a limit point of E n (y i ) for some point y i ∈ Y i so that we have ν i ∈ G i (i = 1, 2, 3). Remark that ν 1 , ν 2 , ν 3 satisfy that min l=1,2 Note that ψdν 1 = 0 < 1 = ψdν 3 so that by Lemma 2.11, I(ψ, f ) = ∅. That is, Then if f satisfies Bowen's specification, by Proposition 21.14 in [12] there is some x ∈ X such that M x (f ) = K(for the case of g−almost product property, it is not difficult to prove just by little modification to the proof of Proposition 21.14 in [12], here we omit the details). That So by Lemma 2.10, x ∈ ∩ k j=1 I(φ j , f ). Notice that Thus by Lemma 2.10 x ∈ R(ψ, f ). We complete the proof.
Theorem 3.1 is to study irregular-mix-regular set under two observable continuous functions. Furthermore, another interesting question is to study the irregular-mix-regular set which is the intersection of several φ−irregular sets and ψ−regular sets. Similar as the statements of Question 1.3, we ask: Question 3.6. Let f be a continuous map of a compact metric space X with g−almost product property. Then for any subsets D 1 , D 2 ⊆ C 0 (X), whether the irregular-mixregular set ( either is empty or has full topological pressure?
If D 1 = D and D 2 is composed of a constant function, then So if Question 3.6 is true, then so is Question 1.3. For systems with g−almost product property and uniform separation property, we have a positive answer of Question 3.6 which particularly implies Theorem 3.1.
Theorem 3.7. Let f be a continuous map of a compact metric space X with g−almost product property and uniform separation property. Then for any subsets D 1 , D 2 ⊆ C 0 (X), the irregular-mix-regular set either is empty or has full topological pressure. We try to state the result of Theorem 3.7 for more general topological dynamical systems.
Theorem 3.9. Let f be a continuous map of a compact metric space X. If f is saturated and the entropy function h · (f ) : M f (X) → R, µ → h µ (f ) is upper continuous, then for any subsets D 1 , D 2 ⊆ C 0 (X), the irregular-mix-regular set either is empty or carries full topological pressure, that is, for any ϕ ∈ C 0 (X).
Fix ǫ > 0 and ϕ ∈ C 0 (X). Take the same ergodic measure µ such that and take θ ∈ (0, 1) close to 1 such that and (1 − θ) ϕ < ǫ, where ϕ = max x∈X |ϕ(x)|. By Lemma 2.9, for any φ i , one can take an invariant measure µ ′ i such that Then take an increasing sequence of numbers θ i ∈ (θ, 1) ↑ 1 such that the invariant measure µ i : Remark that every µ i satisfies φdµ = φdµ i . Let Since µ i converges to µ, it is easy to check that K is connected and compact and every measure ν ∈ K satisfies that h ν (f ) + ϕdν > P (X, ϕ, f ) − 2ǫ.
Since f is saturated, then for above K one has So the left to end the proof is only to show Then there are two sequences of n j , m j ↑ +∞ such that in weak * topology lim j→∞ 1 n j This implies x ∈ I(φ i , f ).

Proof of Theorem 1.9
Remark that Theorem 1.9 is a generalization of Theorem 1.1. Here we need to consider multiple observable functions different from [35,36] which only considers one function. But the idea is just to adapt the proof of [35,36](Remark that the result of [36] is for the case of topological entropy but by slight modification its idea is still valid for topological pressure). So we only give a sketch of the proof and omit the details. Assume k j=1 I(φ j , f ) = ∅.
For a fixed 1 ≤ j ≤ k, obviously the solutions (λ 1 , · · · , λ k ) of linear equation k i=1 ( φ j dµ 1,i − φ j dµ 2,i )λ i = 0 form a k − 1 dimensional closed linear subspace of R k , denoted by P j . Notice that 1≤j≤k P j is the union of finite k − 1 dimensional closed linear subspaces so that is open and dense in R k . One can take k positive numbers of λ 1 , · · · , λ k such that for any For example, if k = 3, (λ 1 , λ 2 , λ 3 ) is chosen in the first octant of R 3 . Let . Then all θ j are positive and k j=1 θ j = 1. If we define θ i µ l,i , l = 1, 2, then for any 1 ≤ j ≤ k, Fix ǫ > 0 and ϕ ∈ C 0 (X). By classical Variational Principle, we can take an ergodic measure µ such that h µ (f ) + ϕdµ > P (X, ϕ, f ) − ǫ.
By [25], when f has the almost specification, we can find two sequence of ergodic measures ν l,i ∈ M f (X) such that h ν l,i (f ) → h ν l , and ν l,i → ν l (l = 1, 2) in weak * topology. Therefore, we can take two measures belonging to these two sequence which we called ρ 1 and ρ 2 respectively such that h ρ l (f ) + ϕdρ l > P (X, ϕ, f ) − 2ǫ (l = 1, 2) and for any 1 ≤ j ≤ k, This is the first step of [35,36] to choose two good ergodic measures but crucial because the ergodicity is important in the proof of [35,36] to avoid the use of uniform separation.
Then we can follow the proof of [35,36] to complete the proof. Roughly speaking, using the above two ergodic measures to construct a set F ⊆ ∩ k j=1 I(φ j , f ) such that the topological pressure of F is larger than P (X, ϕ, f ) − 2ǫ. In this process, Entropy Distribution Principle plays an important role. One can see [35,36] for more details.

Simple Applications
In this section we compute a detailed example as a simple application.
Let f : S 1 → S 1 , x → 2x mod 1. It is known that f is expansive and satisfies Bowen's specification(it is also known f is topologically conjugated to the shift of two symbols except at countably many points). Consider D is a particular subset of C 0 (S 1 ) consisted of some sine functions and cosine functions from the basis of Fourier series. and it is easy to check every function in D is trulyobservable. In fact, let µ be the Dirac measure supported on the fixed point 0 ∈ S 1 and let µ n be the periodic measure supported on the periodic orbit which is the ω−limit set of the orbit , · · · , 2 k 7n , · · · } mod 1.