Completely-Irregular Set in Dynamics

• The completely-irregular set is defined as the collection of points with nonconvergent Birkhoff averages for every observable showing irregular behavior.
• It is characterized by residuality, full topological pressure, and dense orbits under assumptions like non-unique ergodicity and the shadowing property.
• Mechanisms such as specification-type gluing and orbit shadowing ensure that these irregular points are topologically large despite being measure-zero.

In topological dynamics, the completely-irregular set usually denotes the set of points whose Birkhoff averages fail to converge for every continuous observable for which irregular behavior occurs somewhere in the system. For a compact metric space $(X,d)$ and a continuous map $T:X\to X$, if

$$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$

and

$$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$

then

$CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$

Xueting Tian uses the closely related notation

$$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$

so that a point in $CI$ is irregular for every truly-observable function (Tian, 2014, Carvalho et al., 2023).

1. Formal definition and conceptual position

For a continuous observable $\varphi\in C^0(X,\mathbb R)$, the Birkhoff average along the orbit of $x$ is

$$A_n(\varphi,x)=\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x)).$$

A point is $T:X\to X$0-regular if this limit exists, and $T:X\to X$1-irregular if it does not. The $T:X\to X$2-irregular set is denoted $T:X\to X$3, and irregularity is also described as “historic behavior” (Carvalho et al., 2023).

The completely-irregular set is an intersection-type object. It is smaller than a single $T:X\to X$4, because it requires simultaneous irregularity for all observables with nonempty irregular set. In Tian’s formulation, this is the intersection over all truly-observable functions; in Carvalho–Coelho–Salgado’s formulation, it is the intersection over $T:X\to X$5, the family of continuous potentials whose irregular set is nonempty (Tian, 2014, Carvalho et al., 2023).

Two related constructions appear repeatedly in the same circle of ideas. The jointly-irregular set is

$T:X\to X$6

for a specified collection $T:X\to X$7, and irregular-mix-regular sets are intersections of some $T:X\to X$8-irregular sets with some $T:X\to X$9-regular sets. These sets are used to interpolate between single-observable irregularity and complete irregularity (Tian, 2014).

2. Existence and the role of ergodicity assumptions

Non-unique ergodicity is a decisive hypothesis. If a system is uniquely ergodic, then all Birkhoff averages converge and no irregular points exist for any $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$0, so $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$1. Conversely, Tian proves that if $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$2 is not uniquely ergodic and has the $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$3-almost product property together with the uniform separation property, then $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$4 is nonempty and in fact residual, hence a dense $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$5 set (Tian, 2014).

A distinct but closely related framework is developed by M. Carvalho, V. Coelho, and L. Salgado. Let $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$6 be a compact metric space without isolated points, and let $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$7 be continuous, transitive, not uniquely ergodic, and endowed with the shadowing property. Under these assumptions, they prove that $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$8, $$I(T,\varphi)=\left\{x\in X:\left(\frac1n\sum_{j=0}^{n-1}\varphi(T^j(x))\right)_{n\in\mathbb N}\ \text{does not converge}\right\},$$9 has positive topological entropy, the measure center satisfies $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$0, the union of basins of ergodic measures is dense in $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$1, and the completely irregular set $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$2 is Baire generic in $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$3 (Carvalho et al., 2023).

In the same setting, transitivity may be read as the existence of a dense orbit, since on compact metric spaces without isolated points this is equivalent to the usual open-set definition of transitivity. For expansive homeomorphisms with the shadowing property, the cited paper further states that positive topological entropy, non-unique ergodicity, and infinite non-wandering set are equivalent to Baire genericity of the completely irregular set (Carvalho et al., 2023).

3. Pressure, category, and orbit-density properties

The strongest global size statement in the cited literature is Tian’s full-pressure theorem. If $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$4 is continuous on a compact metric space, has the $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$5-almost product property and uniform separation, and is not uniquely ergodic, then for every $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$6,

$$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$7

Thus the completely-irregular set carries full topological pressure. The same work states that $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$8 is residual in the support of all invariant measures, and that this full-pressure phenomenon extends the finite-observable jointly-irregular theory (Tian, 2014).

This topological largeness coexists with measure-theoretic smallness. The same source states that the completely-irregular set is as large as possible from the topological perspective, despite being negligible from the measure-theoretic perspective, since any invariant measure gives it measure zero (Tian, 2014).

In the shadowing framework, Carvalho–Coelho–Salgado prove a complementary largeness result in the sense of Baire category: $$H(X,T)=\left\{\varphi\in C^0(X,\mathbb R): I(T,\varphi)\neq\emptyset\right\},$$9 is Baire generic. They also prove a strong orbit-theoretic property: every completely irregular point has a dense orbit, equivalently

$CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$0

Their theorem therefore does not merely identify a large exceptional set; it identifies a generic set of points whose orbit statistics never stabilize for any potential in $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$1, while those orbits are still dense in the whole phase space (Carvalho et al., 2023).

Framework	Hypotheses	Conclusion for the completely-irregular set
Tian	$CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$2-almost product, uniform separation, not uniquely ergodic	nonempty, residual, full topological pressure
Carvalho–Coelho–Salgado	compact metric space without isolated points, continuous transitive map, shadowing, not uniquely ergodic	Baire generic; every completely irregular point has dense orbit

4. Dynamical mechanisms: specification-type gluing and shadowing

The principal construction mechanisms are specification-like orbit concatenation, shadowing, and variational arguments. Tian’s analysis uses the $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$3-almost product property and uniform separation to obtain variational-principle-type control on saturated sets, together with entropy-dense ergodic measures and residuality arguments. In this framework, one constructs orbits with prescribed empirical behavior so that Birkhoff averages fail to converge simultaneously for many observables (Tian, 2014).

Carvalho–Coelho–Salgado isolate the shadowing property as a sufficient mechanism in a transitive, non-uniquely ergodic setting. The shadowing property says that for every $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$4 there exists $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$5 such that every $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$6-pseudo-orbit $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$7, satisfying $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$8, can be $CI(X,T):=\bigcap_{\varphi\in H(X,T)} I(T,\varphi).$9-shadowed by a true orbit: there exists $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$0 such that $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$1 for all $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$2. Their results show that transitivity plus shadowing already force a generic completely-irregular set, without an explicit specification assumption (Carvalho et al., 2023).

Single-observable irregular-set papers clarify why these mechanisms are effective. Chunlin Liu and Xue Liu show that the almost weak specification property is essential in constructing large Moran-like fractal sets within the irregular set, by repeatedly concatenating pieces of orbits corresponding to different invariant measures. The gluing requires shadowing prescribed orbit segments up to small error and up to negligible gaps, which is precisely what almost weak specification grants via tempered gap functions (Liu et al., 2022). This suggests that the completely-irregular theory inherits its technical architecture from the broader specification-and-shadowing program for historic behavior.

A related fibered version appears for skew product transformations driven by a uniquely ergodic base and Anosov, topologically mixing fibers. There the orbit-gluing technique is supplied by the fiber specification property, and if the irregular set of a continuous observable is nonempty then it is nonempty, residual, and carries full fiber topological entropy on $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$3-almost every fiber (Liu et al., 2024). The cited work does not treat the completely-irregular set explicitly, but it identifies the same orbit-gluing paradigm in a non-product setting.

5. Terminological variants and neighboring notions

The term “completely-irregular set” is not uniform across the cited literature. In beta-expansion dynamics, the relevant object is

$$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$4

where

$$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$5

Here $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$6 is the set of points for which the limit of $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$7 does not exist. The cited paper proves that $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$8 is residual for all $$\hat C^0(X):=\{\varphi\in C^0(X): I(\varphi,f)\neq\varnothing\},\qquad CI(f):=\bigcap_{\varphi\in \hat C^0(X)} I(\varphi,f),$$9, and that

$CI$0

In the same setting, the extremely irregular set

$CI$1

is residual, has box and packing dimension $CI$2, and Hausdorff dimension $CI$3 (Zheng et al., 2016).

A different terminological choice appears in the study of topologically mixing subshifts of finite type. There, the details identify the “completely-irregular set” as

$CI$4

the collection of points irregular for some $CI$5. Under the assumption of at least two safe symbols, there exists an uncountable collection of pairwise disjoint, dense, invariant sets inside an irregular set for a continuous observable, each with full topological entropy and full Hausdorff dimension (Burgos, 2024). This usage differs from the intersection-based definition in Tian and in Carvalho–Coelho–Salgado.

These differences matter conceptually. In one line of work, “completely-irregular” means irregular for every relevant observable; in another, it refers to points whose limit fails to exist for a specific non-Birkhoff quantity; and in another, it is attached to a union over observables. The cited works therefore document a genuine terminological variation rather than a single universal definition (Zheng et al., 2016, Burgos, 2024).

6. Relation to single-observable irregular sets and metric mean dimension

A substantial part of the recent literature studies $CI$6 for a fixed continuous observable rather than the completely-irregular intersection. Liu and Liu prove that if $CI$7 has the almost weak specification property and $CI$8 is continuous, then the irregular set $CI$9 is either empty or carries full Bowen upper and lower metric mean dimension, provided the corresponding upper or lower classical metric mean dimension of the whole system is finite and positive (Liu et al., 2022).

An analogous result holds for metric mean dimension with potential. If $\varphi\in C^0(X,\mathbb R)$0 has the specification property and $\varphi\in C^0(X,\mathbb R)$1 is continuous, then the multifractal irregular set

$\varphi\in C^0(X,\mathbb R)$2

is either empty or carries full Bowen upper and lower metric mean dimension with potential (Zhang et al., 2024).

For the topic of completely-irregular sets, the limitation is explicit in one of the principal sources: the paper of Liu and Liu states that there is no direct discussion or theorem on the completely-irregular set, and that any statements about it would require additional arguments (Liu et al., 2022). By contrast, the metric-mean-dimension-with-potential paper states that, while it focuses on $\varphi\in C^0(X,\mathbb R)$3 for a single potential, the construction can be adapted, in principle, to the completely-irregular set under strong specification-type properties by choosing countable dense families of potentials and carrying out a diagonal construction (Zhang et al., 2024).

The present state of the cited results therefore separates two levels of theory. For complete irregularity itself, the strongest direct theorems are formulated in terms of Baire genericity, residuality, dense $\varphi\in C^0(X,\mathbb R)$4 structure, full topological pressure, and dense orbits (Tian, 2014, Carvalho et al., 2023). For metric mean dimension, the direct theorems currently cited are single-observable statements, with the completely-irregular case left as an extension requiring further work (Liu et al., 2022, Zhang et al., 2024).