2.1 Dynamical systems (standing assumptions)

We assume that X is a compact metric space, $\rho $ is a metric for X, and $T\colon X\to X$ is a continuous map. All results remain the same if $\rho $ is replaced by another compatible metric. Given $x\in X$ , we distinguish between the orbit of x, which is a set $\{T^n(x):n\ge 0\}\subset X$ , and the trajectory of x, which is a sequence $\underline {x}_T=(T^j(x))_{j=0}^{\infty} \in X^{\infty} $ .

2.2 Note on invertibility

Observe that by default we consider transformations that are not necessarily invertible, but some of the results we invoke from the literature assume invertibility. It is easy to see that in all such cases, it is enough to apply the theorem we need to use the natural extension of non-invertible transformations.

2.3 Measure-preserving systems

Most of the standard texts on ergodic theory work with measure-preserving transformations of standard Lebesgue spaces. The latter are measure spaces arising as completions of probability measures on Polish metric spaces endowed with their Borel $\sigma $ -algebras. In this approach, it is hard to consider different measures on the same underlying space, since the $\sigma $ -algebra depends non-trivially on the measure. This is the primary reason why we work in the Borel category. If necessary, when we work with a single invariant Borel measure $\mu $ , we can take the completion of our measure space to obtain a Lebesgue space. All properties of measure-preserving systems considered here remain the same for the original system and its completion, that is, a measure-preserving system $(X,\mathscr {X},\mu ,T)$ has one of these properties if and only if the completed system $(X,\tilde {\mathscr {X}}_\mu ,\tilde \mu ,T)$ has the property.

2.4 Invariant measures and generic sequences

We write $\mathscr {X}$ for the Borel $\sigma $ -algebra of X and ${\mathcal {M}}(X)$ for the set of all Borel probability measures on X. By ${\mathcal {M}_T}(X)$ we denote T-invariant measures in ${\mathcal {M}}(X)$ . We write ${{\mathcal {M}}_T^e}(X)$ for the set of ergodic invariant measures. The quadruple $\mathbf {X}=(X,\mathscr {X},\mu ,T)$ is a measure-preserving system, which is invertible, whenever T is a homeomorphism. We give ${\mathcal {M}}(X)$ the weak $^*$ topology. Recall that $(\mu _n)_{n=1}^{\infty} $ converges to $\mu $ in ${\mathcal {M}}(X)$ if and only if $\int \varphi \,d\mu _n\to \int \varphi \,d\mu $ for every continuous ${\varphi \colon X\to \mathbb {R}}$ . The weak $^*$ topology on ${\mathcal {M}}(X)$ is compact and compatible with the Prokhorov metric

$$ \begin{align*} D_{\mathrm{P}}(\mu,\nu)=\inf\{\varepsilon>0\,:\,\mu(B)\leq\nu(B^{\varepsilon})+\varepsilon\text{ for every Borel set }B\subset X\}, \end{align*} $$

where $B^\varepsilon =\{y\in X\,:\,\operatorname {\mathrm {dist}}(y,B)<\varepsilon \}$ denotes the $\varepsilon $ -hull of B. For $x\in X$ , let $\hat \delta (x)\in \mathcal M(X)$ be the Dirac measure supported on $\{x\}$ . Let ${\mathfrak {m}}(\underline {x},n)$ denote the n-empirical measure of $\underline {x}=(x_j)_{j=0}^{\infty} \in X^{\infty }$ , that is, ${\mathfrak {m}}(\underline {x},n)=(1/n)(\hat \delta (x_0)+\hat \delta (x_1)+\cdots +\hat \delta (x_{n-1}))$ . Given $x\in X$ , we put ${\mathfrak {m}}_T(x, n)={\mathfrak {m}}(\underline {x}_T,n)$ . A measure $\mu \in {\mathcal {M}}(X)$ is generated by $\underline {x}\in X^{\infty} $ if $\mu $ is the limit of some subsequence of $({\mathfrak {m}}(\underline {x},n))_{n=1}^{\infty }$ . The set of all measures generated by $\underline {x}$ is denoted by $V({\underline {x}})$ . We say that $\underline {x}\in X^{\infty} $ is a generic sequence for $\mu $ (respectively, an ergodic sequence) if $V({\underline {x}})=\{\mu \}$ for some (ergodic) $\mu \in {\mathcal {M}}(X)$ . We write $\operatorname {\mathrm {Gen}}(\mu )$ for the set of all sequences in $X^{\infty} $ that are generic for $\mu $ . For $z\in X$ we define $V_T({z})=V({\underline {z}_T})$ , and we call z a generic point (ergodic point) if $\underline {z}_T$ is a generic sequence (respectively, ergodic) sequence. Note that every invariant measure has a generic sequence in $X^{\infty} $ [Reference Kwietniak, Łącka and Oprocha36, Reference Sigmund44], while a non-ergodic invariant measure may have no generic points. Furthermore, one can choose a generic sequence that is a quasi-orbit. A quasi-orbit is built from long pieces of orbits in such a way that the set of positions at which a quasi-orbit switches from one piece of orbit to another has zero asymptotic density.

It is easy to see that every measure generated by a quasi-orbit for T must be T-invariant. Furthermore, the GIKN construction yields a quasi-orbit that is generic for the invariant measure it produces. We work with that quasi-orbit to demonstrate the properties of the underlying measure.

2.5 Symbolic systems

Let $\mathscr {A}$ be a finite set with the discrete topology. We endow $\mathscr {A}^{\kern1.5pt\infty} $ with the product topology and call it the full shift over the alphabet $\mathscr {A}$ . The shift map is the map $\sigma \colon \mathscr {A}^{\kern1.5pt\infty} \to \mathscr {A}^{\kern1.5pt\infty} $ given by $\sigma ((x_n)_{n=0}^{\infty} )=(x_{n+1})_{n=0}^{\infty} $ . The set of (ergodic) shift-invariant measures is denoted by ${\mathcal {M}}_{\sigma }(\mathscr {A}^{\kern1.5pt\infty} )$ ( ${\mathcal {M}}^e_{\sigma }(\mathscr {A}^{\kern1.5pt\infty} )$ ). When ${\mathscr {A}=\{0,1,\ldots ,k-1\}}$ for some $k\in \mathbb {N}$ , we write $\Omega _k=\{0,1,\ldots ,k-1\}^{\infty} $ . We call the elements of $\mathscr {A}^n$ words of length n over $\mathscr {A}$ . Let $\mathscr {A}^+=\bigcup _{n\ge 1}\mathscr {A}^n$ and $|u|$ stand for the length of $u\in \mathscr {A}^+$ . Every word $u\in \mathscr {A}^+$ determines a cylinder set $[u]\subset \mathscr {A}^{\kern1.5pt\infty} $ consisting of all sequences in $\mathscr {A}^{\kern1.5pt\infty} $ whose first $|u|$ symbols coincide with u. Cylinders form a clopen base for the topology of $\mathscr {A}^{\kern1.5pt\infty} $ and generate the Borel $\sigma $ -algebra $\mathscr {B}$ of $\mathscr {A}^{\kern1.5pt\infty} $ . Given two n-words $u=u_0u_1\cdots u_{n-1}$ and $w=w_0w_1\cdots w_{n-1}$ over $\mathscr {A}$ , we define the Hamming distance between u and w as

$$ \begin{align*} \bar{d}_n(u,w)=\frac{1}{n}|\{0\le j<n:u_j\neq w_j\}|. \end{align*} $$

The edit metric between two strings (words) of length n equals $1-k/n$ , where k is the minimum number of symbols that must be removed from each string so that the remaining strings are identical. In other words, the edit distance between u and w is given by

$$ \begin{align*} \bar{f}_n(u,w)=1-\frac{k}{n}, \end{align*} $$

where k is the largest among those integers $\ell $ such that for some ${0\le i_1<i_2<\cdots <i_\ell <n}$ and $0\le j_1<j_2<\cdots <j_\ell <n$ we have $u_{i_s}=w_{j_s}$ for $s=1,\ldots ,\ell $ . For two infinite sequences ${\omega }=\omega _0\omega _1\omega _2\cdots $ , ${\omega }'=\omega ^{\prime }_0\omega ^{\prime }_1\omega ^{\prime }_2\cdots $ in $\mathscr {A}^{\kern1.5pt\infty} $ we set

(1) $$ \begin{align} \bar{d}({\omega},{\omega}') & =\limsup_{n\to\infty}\bar{d}_n({\omega},{\omega}')=\limsup_{n\to\infty} \bar{d}_n(\omega_0\omega_1\cdots \omega_{n-1},\omega^{\prime}_0\omega^{\prime}_1\cdots \omega^{\prime}_{n-1}) \nonumber\\& = \bar{d}(\{j\ge 0:\omega_j\neq\omega^{\prime}_j\}), \end{align} $$

(2) $$ \begin{align}\bar{f}({\omega},{\omega}') & = \limsup_{n\to\infty}\bar{f}_n({\omega},{\omega}')=\limsup_{n\to\infty} \bar{f}_n(\omega_0\omega_1\cdots \omega_{n-1},\omega^{\prime}_0\omega^{\prime}_1\cdots \omega^{\prime}_{n-1}), \end{align} $$

(3) $$ \begin{align} \hat{f}({\omega},{\omega}')& = \inf\{\varepsilon>0:\text{ there are increasing sequences } (i_r), (i_r') \text{ in } \mathbb{N}^{\infty} \nonumber\\&\qquad\quad \text{of lower density at least } 1-\varepsilon \text{ for which } \omega_{i_r}=\omega^{\prime}_{i_r'} \text{ for all } r\ge0 \}. \end{align} $$

The functions $\bar {d}$ , $\bar {f}$ , and $\hat {f}$ are pseudometrics on $\mathscr {A}^{\kern1.5pt\infty} $ . Furthermore, $\bar {f}({\omega },{\omega }')\le \hat {f}({\omega },{\omega }')$ and $\bar {f}({\omega },{\omega }')\le \bar {d}({\omega },{\omega }')$ for $\omega ,\omega '\in \mathscr {A}^{\kern1.5pt\infty} $ , and $\bar {f}$ , $\hat {f}$ are uniformly equivalent pseudometrics on $\mathscr {A}^{\kern1.5pt\infty} $ (see [Reference Ornstein, Rudolph and Weiss39]).

2.6 Processes

We write $\mathbf {P}^m(X)$ for the set of all Borel measurable partitions of X into at most m sets, called atoms. For $\mathcal P\in \mathbf P^k(X)$ we write ${\mathcal {P}}=\{P_0,\ldots ,P_{k-1}\}$ regardless of the actual number of non-empty elements in $\mathcal P$ and we agree that $P_j=\emptyset $ for $|{\mathcal {P}}|\le j<k$ . Let $\mathbf {X}=(X,\mathscr X, \mu , T)$ be a measure-preserving system and let $\mathcal P=\{P_0,P_1,\ldots , P_{k-1}\}\in \mathbf {P}^k(X)$ . We identify $\mathcal P$ with a function $\mathcal P\colon X\to \{0,\ldots , k-1\}$ defined by $\mathcal P(x)=j$ for $x\in P_j$ . The pair $(\mathbf {X},\mathcal P)$ is called a process (see [Reference Glasner27, pp. 273]). A coding of $\underline {x}=(x_j)_{j=0}^{\infty }\in X^{\infty }$ is ${\mathcal {P}}(\underline {x})=(\mathcal P(x_j))_{j=0}^{\infty }\in \Omega _k$ . The map $\underline {{\mathcal {P}}}\colon X\to \Omega _k\text { given by } \underline {{\mathcal {P}}}(x)=\mathcal P(\underline {x}_T)$ defines a homomorphism of $\mathbf {X}$ and $(\Omega _k,\mathscr {B}, \mu _{\mathcal P}, \sigma )$ , where $\mu _{\mathcal P}=\underline {{\mathcal {P}}}_*(\mu )$ is the pushforward of the measure $\mu $ . For $n>0$ and $\underline {x}\in X^{\infty }$ let $\varphi ^n_{\mathcal P}(\underline {x})=\mathcal P(x_0)\mathcal P(x_1)\cdots \mathcal P(x_{n-1})\in \, \{0,1,\ldots , k-1\}^n$ . Let $\mathcal P^n$ be the nth join of $\mathcal P$ given by

$$ \begin{align*} \mathcal P^n=\bigvee_{j=0}^{n-1}T^{-j}(\mathcal P)=\{P_{i_0}\cap T^{-1}(P_{i_1})\cap\cdots\cap T^{-n+1}(P_{i_{n-1}}):P_{i_j}\in\mathcal P\text{ for }0\le j<n\}. \end{align*} $$

Note that for $x\in X$ we may write $\varphi ^n_{\mathcal P}(\underline {x}_T)=\mathcal P^n(x)$ , because $\varphi ^n_{\mathcal P}(\underline {x}_T)=i_0i_1\cdots i_{n-1}$ if and only if ${\mathcal {P}}^n(x)=P_{i_0}\cap T^{-1}(P_{i_1})\cap \cdots \cap T^{-n+1}(P_{i_{n-1}})$ . The measure-preserving system $(\Omega _k,\mathscr {B}, \mu _{\mathcal P}, \sigma )$ is called the symbolic representation of $\mathbf {X}$ with respect to the partition $\mathcal P$ , and $\mu _{\mathcal P}$ is the symbolic representation measure of $\mu $ . We endow $\mathbf P^k(X)$ with the distance $d_1^\mu $ given for ${\mathcal {P}},\mathcal {Q}\in \mathbf {P}^k(X)$ by

$$ \begin{align*} d^\mu_1({\mathcal{P}},\mathcal{Q})= \dfrac{1}{2}\kern-1pt \sum_{j=0}^{k-1} \mu(P_j \div Q_{j})\kern-1pt =\kern-1pt \dfrac{1}{2}\kern-1pt \sum_{j=0}^{k-1}\kern-1pt \int_X\!|\chi_{P_j}\kern-1pt -\kern-1pt \chi_{Q_j}|\,\text{d}\mu\kern-1pt =\kern-1pt \mu(\{x\in X:{\mathcal{P}}(x)\neq\mathcal{Q}(x)\}). \end{align*} $$

Note that the definition of $d^\mu _1$ takes into account the order of the partition’s elements. We tacitly identify Borel partitions ${\mathcal {P}},\mathcal {Q}\in \mathbf {P}^k(X)$ with $d^\mu _1({\mathcal {P}},\mathcal {Q})=0$ . With this identification $d^\mu _1$ is a complete metric for $\mathbf {P}^k(X)$ .

2.7 Entropy

For a finite measurable partition $\mathcal P$ and $\mu \in {\mathcal {M}}_T(X)$ we denote by $h(\mu , \mathcal P)$ the entropy of $\mathcal P$ with respect to $\mu $ and T and by $h(\mu )$ the entropy of $\mu $ with respect to T, that is, $h(\mu )=\sup _{\mathcal P}h(\mu , \mathcal P)$ , where $h(\mu , \mathcal P)=\inf _{n\in \mathbb {N}}-\sum _{P\in \mathcal P^n}\mu (P)\log \mu (P)$ . The real-valued function $\mathcal P\mapsto h(\mu ,\mathcal P)$ is uniformly continuous on $\mathbf {P}^k(X)$ equipped with $d^\mu _1$ [Reference Glasner27, Lemma 15.9 (5)].

2.8 Faithful coding

For ${\mathcal {P}}\in \mathbf {P}^k(X)$ we define $\partial {\mathcal {P}}=\partial P_0\cup \cdots \cup \partial P_{k-1}$ . A partition ${\mathcal {P}}\in \mathbf {P}^k(X)$ with $\mu (\partial {\mathcal {P}})=0$ is called faithful for $\mathbf {X}$ .

Proof. Note the following two properties of the boundary operator $\partial $ : $\partial (Y\cap Z)\subset \partial Y\cup \partial Z$ for $Y,Z\subset X$ and $\partial T^{-1}(U)\subset T^{-1}(\partial U)$ for any $U\subset X$ . Using this and $\mu (\partial P_j)=0$ for every $0\le j < k$ , we see that

$$ \begin{align*} \mu\bigg(\partial\bigg(\bigcap_{i=0}^{m-1}T^{-i}(P_{j_i})\bigg)\bigg)=0\quad\text{for every }m\ge 1 \text{ and }0\le j_0, j_1,\ldots,j_{m-1}< k. \end{align*} $$

In other words, $\mu (\partial {\mathcal {P}}^m)=0$ for all $m\ge 1$ . Then for every $m\ge 1$ and $0\le j_0,j_1,\ldots , j_{m-1}< k$ we have

$$ \begin{align*} \lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\chi_A(x_j)=\mu(A)\quad \text{for }A=\bigcap_{i=0}^{m-1}T^{-i}(P_{j_i})\in{\mathcal{P}}^m. \end{align*} $$

Note that $\omega ={\mathcal {P}}(\underline {x})$ is an orbit for $\sigma $ and observe that $\omega $ is generic for a $\sigma $ -invariant measure $\mu '$ such that

$$ \begin{align*} & \mu'([j_0j_1\cdots j_{m-1}])=\mu\bigg(\bigcap_{i=0}^{m-1}T^{-i}(P_{j_i})\bigg)\\ & \quad\text{for every }m\ge 1 \text{ and } 0\le j_0,j_1,\ldots,j_{m-1}< k. \end{align*} $$

Hence, $\mu '$ and $\mu _{{\mathcal {P}}}$ agrees on cylinders in $\Omega _k$ . This implies $\mu '=\mu _{{\mathcal {P}}}$ .

2.9 Kakutani equivalence

Kakutani equivalence serves as a natural equivalence relation among transformations that preserve an ergodic non-atomic measure, weaker than the conventional concept of isomorphism (however, in some sense equally hard; cf. [Reference Gerber and Kunde26]). Although entropy does not remain invariant under Kakutani equivalence, Abramov’s formula implies that this relation retains the categories of zero, positive and finite, as well as infinite entropy transformations. Let $\mathbf {X}=(X,\mathscr X, \mu , T)$ be a measure-preserving system. For a set $E\in \mathscr X$ with $\mu (E)>0$ and $x\in E$ we define the return time $n(x)=\inf \{k>0: T^k(x)\in E\}$ . This function is finite for $\mu $ -almost every $x\in E$ , and we define the induced transformation $T_E\colon E\to E$ by $T_E(x)=T^{n(x)}(x)$ . Measure-preserving systems $\mathbf {X}=(X,\mathscr X, \mu , T)$ and $\mathbf Y=(Y,\mathscr Y, \nu , S)$ are Kakutani equivalent (recall that Katok calls this relation monotone equivalence) if there exist $E\in \mathscr X$ with $\mu (E)>0$ and $F\in \mathscr Y$ with $\nu (F)>0$ such that $T_E$ is isomorphic with $S_F$ .

2.10 Loosely Bernoulli systems

Feldman [Reference Feldman21] studied the isomorphism problem in ergodic theory. He introduced a new property (called loose Bernoulliness) for finite partitions of a measure-preserving system $(X,B,\mu ,T)$ . He used it to construct important examples of K-automorphisms which are loosely Bernoulli but not Bernoulli. In particular, these measure-preserving systems have positive entropy. The definition of loosely Bernoulli partition follows Ornstein’s definition [Reference Ornstein38] of very weak Bernoulli partition with the Hamming distance of strings of symbols of length n replaced by the weaker edit metric $\bar {f}_n$ . Feldman’s idea was subsequently extended by Ornstein, Rudolph and Weiss [Reference Ornstein, Rudolph and Weiss39]. It turns out that, in each of three entropy classes, the Kakutani equivalence class of loosely Bernoulli transformations is the simplest one (see [Reference Ornstein, Rudolph and Weiss39] for more details). Positive and finite (respectively, infinite) entropy transformations Kakutani equivalent to a Bernoulli shift coincide with loosely Bernoulli and positive and finite (respectively, infinite) entropy transformations. Zero-entropy loosely Bernoulli transformations form the Kakutani equivalence class of any ergodic rotation of a compact infinite group (Kronecker system). According to Feldman and Nadler [Reference Feldman and Nadler22], members of the latter equivalence class are called loosely Kronecker following a suggestion of Marina Ratner. We adapt this terminology and provide a characterization of these systems in §2.13 (cf. Theorem 9.2).

2.11 Feldman’s $\bar {f}$ and Ornstein’s $\bar {d}$ metrics for shift-invariant measures on $\mathscr {A}^{\kern1.5pt\infty} $

Let $\mu $ and $\mu '$ be ergodic shift-invariant measures on $\mathscr {A}^{\kern1.5pt\infty} $ . By $\mu _n$ (respectively, $\mu ^{\prime }_n$ ) we denote the restriction of $\mu $ (respectively, $\mu '$ ) to the set of all n-cylinders, that is, these are measures that $\mu $ (respectively, $\mu '$ ) defines on $\mathscr {A}^n$ via the projections onto first n coordinates. A joining of $\mu $ and $\mu '$ is any $\sigma \times \sigma $ -invariant measure on $\mathscr {A}^{\kern1.5pt\infty} \times \mathscr {A}^{\kern1.5pt\infty} $ whose marginals are $\mu $ and $\mu '$ . We write $J(\mu ,\mu ')$ for the set of all joinings of $\mu $ and $\mu '$ . Similarly, $J_n(\mu ,\mu ')$ denotes the set of all measures $\unicode{x3bb} _n$ on $\mathscr {A}^n\times \mathscr {A}^n$ whose marginals are $\mu _n$ and $\mu ^{\prime }_n$ .

Define

(4)

(5) $$ \begin{align} \bar{f}(\mu,\mu') & = \limsup_{n\to\infty}\bar{f}_n(\mu,\mu'). \end{align} $$

One can prove (see [Reference Ornstein, Rudolph and Weiss39]) that (5) defines a distance between measures $\mu $ and $\mu '$ on $\mathscr {A}^{\kern1.5pt\infty} $ known as the $\bar {f}$ -metric. Ornstein’s $\bar {d}$ -metric on ${\mathcal {M}}_\sigma (\mathscr {A}^{\kern1.5pt\infty} )$ is defined analogously with the Hamming distance $\bar {d}_n$ on $\mathscr {A}^n$ replacing the edit distance $\bar {f}_n$ in (4) (see [Reference Shields43]).

2.12 Properties of $\bar {f}$

The following result is a direct corollary of [Reference Ornstein, Rudolph and Weiss39].

Note also that the entropy function $\mu \mapsto h(\mu )$ is uniformly continuous with respect to the $\bar {f}$ -metric on the space of shift-invariant measures on $\mathscr {A}^{\kern1.5pt\infty} $ (for ergodic measures this is [Reference Ornstein, Rudolph and Weiss39, Proposition 3.4]; the assumption of ergodicity can be removed thanks to the main result of [Reference Downarowicz, Kwietniak and Łącka20]).

2.13 Loosely Kronecker systems

An ergodic measure-preserving system $\mathbf {X}=(X,\mathscr {X},\mu ,T)$ is loosely Kronecker if it has zero entropy and for every finite Borel partition ${\mathcal {P}}$ of X and every $\varepsilon>0$ there are $n>0$ and a set $A_n$ of atoms of ${\mathcal {P}}^n=\bigvee _{j=0}^{n-1}T^{-j}({\mathcal {P}})$ such that $\mu (A_n)>1-\varepsilon $ and $\bar {f}_n(u,w)<\varepsilon $ for $u,w\in A_n$ (here, as usual, we identify atoms of the partition ${\mathcal {P}}^n$ with words of length n over the alphabet $\{0,1,\ldots ,|{\mathcal {P}}|-1\}$ ).

3.1 Topological backbone of the GIKN construction

In this subsection, we sketch the main features of the GIKN construction. Our results about $\overline {fk}$ -limits like Theorems 8.1 and 9.1 apply to any measure defined in this way, because the GIKN construction leads to $\overline {fk}$ -convergent sequences of periodic orbits (see Theorem 6.2).

Definition 3.1. We say that a T-periodic orbit $\Gamma $ is a $(\gamma ,\kappa )$ -good approximation of a T-periodic orbit $\Lambda $ if there are a subset $\Delta $ of $\Gamma $ with $|\Delta |/|\Gamma |\ge \kappa $ and a constant-to-one surjection $\psi \colon \Delta \to \Lambda $ (called a $(\gamma ,\kappa )$ -projection) such that for each $y\in \Delta $ and ${0\le j <|\Lambda |}$ we have

$$ \begin{align*} \rho(T^j(y),T^j(\psi(y)))<\gamma. \end{align*} $$

The following is a slightly reformulated version of [Reference Bonatti, Díaz and Gorodetski8, Lemma 2.5]. The proof that $\mu $ is ergodic in [Reference Bonatti, Díaz and Gorodetski8] invokes [Reference Gorodetskiĭ, Ilyashenko, Kleptsyn and Nalskiĭ29, Lemma 2].

Theorem 3.3. Let $(\Gamma _n)_{n\in \mathbb {N}}$ be a GIKN sequence of T-periodic orbits. If $(\mu _n)_{n=1}^{\infty} $ is the associated sequence of ergodic measures, then $(\mu _n)_{n\in \mathbb {N}}$ weak $^*$ converges to an ergodic measure $\mu $ supported on the topological limit of $(\Gamma _n)_{n\in \mathbb {N}}$ , that is,

$$ \begin{align*} \operatorname{\mathrm{supp}}\mu=\bigcap_{k=1}^{\infty}\overline{\bigcup_{n\ge k}\Gamma_n}. \end{align*} $$

With this terminology, Theorem 3.3 states that every GIKN sequence determines a GIKN measure.

3.2 Lyapunov exponents

We discuss how the GIKN construction is used to find non-hyperbolic measures. This part is logically independent of the rest of the paper.

Let M be a smooth Riemannian manifold with $\dim M=m$ . If $f\colon M\to M$ is a diffeomorphism and $\mu $ is an ergodic f-invariant measure, then there exist a set $\Lambda \subset M$ of full $\mu $ -measure and real numbers $\chi ^1_\mu \le \cdots \le \chi ^m_\mu $ such that for every $x\in \Lambda $ and non-zero vector $v\in T_xM$ one has

$$ \begin{align*} \lim_{n\to\infty}\frac{1}{n}\log\|Df_x^n(v)\|=\chi^i_\mu\quad\text{for some }i=1,\ldots,m. \end{align*} $$

The number $\chi ^i_\mu $ is the ith Lyapunov exponent of the measure $\mu $ . If there exist a closed f-invariant set $\Xi \subset M$ and a continuous $Df$ -invariant direction field $\mathcal E=(E_x)_{x\in \Xi }\subset T_\Xi M$ with $\dim E_x=1$ for $x\in \Xi $ , then for every measure $\nu \in {\mathcal {M}}_f(M)$ with $\operatorname {\mathrm {supp}}\nu \subset \Xi $ there is a Lyapunov exponent $\chi ^{\mathcal E}(\nu )$ of $\nu $ associated with $\mathcal E$ in the following sense: for $\nu $ -almost every $x\in M$ and a non-zero vector $v\in E_x$ one has

$$ \begin{align*} \lim_{n\to\infty}\frac{1}{n}\log\|Df_x^n(v)\|=\chi^{\mathcal E}(\nu). \end{align*} $$

Furthermore, if $(\mu _n)_{n\in \mathbb {N}}$ is a sequence in ${\mathcal {M}}^e_f(\Xi )$ , $\mu \in {\mathcal {M}}^e_f(M)$ and $\mu _n\to \mu $ as $n\to \infty $ in the weak $^*$ topology (this implies that $\operatorname {\mathrm {supp}}\mu \subset \Xi $ as well), then $\chi ^{\mathcal E}(\mu _n)\to \chi ^{\mathcal E}(\mu )$ as $n\to \infty $ [Reference Gorodetskiĭ, Ilyashenko, Kleptsyn and Nalskiĭ29, Lemma 1].

4.1 Besicovitch pseudometric $D_{\mathrm {B}}$

For $\underline {x}=(x_j)_{j=0}^{\infty} ,\,\underline {z}=(z_j)_{j=0}^{\infty} \in X^{\infty }$ we define the Besicovitch pseudometric $D_{\mathrm {B}}$ on $X^{\infty} $ as

$$ \begin{align*} D_{\mathrm{B}}(\underline{x}, \underline{z})=\limsup\limits_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\rho(x_j, z_j). \end{align*} $$

It is known [Reference Kwietniak, Łącka and Oprocha36] that $D_{\mathrm {B}}$ is uniformly equivalent to $D_{\mathrm {B}}'$ given by

(6) $$ \begin{align} D^{\prime}_{\mathrm{B}}(\underline{x},\underline{z})=\inf \{\delta>0: \bar{d}(\{n\ge 0: \rho(x_n,z_n)\ge \delta\})<\delta\}. \end{align} $$

Given $T\colon X\to X$ , the Besicovitch pseudometric $D_{\mathrm {B}}$ on X is defined by $D_{\mathrm {B}}(x,y)=D_{\mathrm {B}}(\underline {x}_T,\underline {z}_T)$ . If $X=\mathscr {A}^{\kern1.5pt\infty} $ and $\rho $ is any metric compatible with the topology on $\mathscr {A}^{\kern1.5pt\infty} $ , then the Besicovitch pseudometric $D_{\mathrm {B}}$ and the $\bar {d}$ -pseudometric are uniformly equivalent on $\mathscr {A}^{\kern1.5pt\infty} $ .

4.2 Pseudometric $\overline {fk}$

Fix $\underline {x}=(x_j)_{j=0}^{\infty} $ , $\underline {z}=(z_j)_{j=0}^{\infty} \in X^{\infty }$ , $\delta>0$ , and $n\in \mathbb {N}$ .

Definition 4.2. An $(n,\delta )$ -match is maximal if its fit is the largest possible. If there is no $(n,\delta )$ -match $\pi $ with $|\pi |\ge 1$ , then the empty match $\pi _\emptyset $ with $|\pi _\emptyset |=0$ is the maximal one. The $(n,\delta )$ -gap between $\underline {x}$ and $\underline {z}$ is given by

$$ \begin{align*} \bar{f}_{n,\delta}(\underline{x},\underline{z})=1-\frac{\max\{|\pi|: \pi \text{ is an } (n,\delta)\text{-match of } \underline{x} \text{ with } \underline{z}\}}{n}. \end{align*} $$

Note that if $X=\mathscr {A}^{\kern1.5pt\infty} $ for some finite alphabet $\mathscr {A}$ and we endow $X=\mathscr {A}^{\kern1.5pt\infty} $ with the standard metric given by

$$ \begin{align*} \rho(\underline{\omega},\underline{\omega}')=\begin{cases} 0 & \mbox{if }\underline{\omega}=\underline{\omega}', \\ 2^{-\min\{j\ge0: \omega_j\neq\omega^{\prime}_j\}} & \mbox{otherwise}, \end{cases} \end{align*} $$

then (with a minor abuse of notation) we have $\bar {f}_{n,1}(\underline {x},\underline {z})=\bar {f}_n(x_0x_1\cdots x_{n-1},z_0z_1\cdots z_{n-1})$ for any $\underline {x},\underline {z}\in \mathscr {A}^{\kern1.5pt\infty} $ .

We note some properties of the $(n,\delta )$ -gap function $\bar {f}_{n,\delta }$ that hold for any $\underline {x}$ , $\underline {z}\in X^{\infty }$ , $\varepsilon ,\delta>0$ , and $n\in \mathbb {N}$ . The proofs are left to the reader.

Definition 4.6. The $\bar {f}_\delta $ -pseudodistance between $\underline {x}$ and $\underline {z}$ is given by

$$ \begin{align*} \bar{f}_{\delta}(\underline{x},\underline{z})=\limsup_{n\to\infty}\bar{f}_{n,\delta}(\underline{x},\underline{z}). \end{align*} $$

Fact 4.9. If $\underline {x}$ and $\underline {z}$ are periodic sequences with a common period N, then

$$ \begin{align*} \bar{f}_{\delta}(\underline{x},\underline{z})=\inf_{n\in\mathbb{N}}\bar{f}_{nN,\delta}(\underline{x},\underline{z})=\lim_{n\to\infty}\bar{f}_{nN+q,\delta}(\underline{x},\underline{z}) \end{align*} $$

for every $0\le q<N$ .

Definition 4.10. The Feldman–Katok pseudometric on $X^{\infty }$ is given by

$$ \begin{align*} \overline{fk}(\underline{x},\underline{z})=\inf\{\delta>0:\bar{f}_{\delta}(\underline{x},\underline{z})<\delta\}. \end{align*} $$

The Feldman–Katok pseudometric on X, also denoted by $\overline {fk}$ , is defined for $x,z\in X$ by

$$ \begin{align*} \overline{fk}(x,z)=\overline{fk}(\underline{x}_T,\underline{z}_T). \end{align*} $$

The symbols $\bar {f}_{n,\delta }(x,z)$ and $\bar {f}_{\delta }(x,z)$ have the obvious meaning.

Proof. It is enough to prove that $\bar {f}_\delta (x,y)=\bar {f}_\delta (x',y')$ for every $\delta>0$ . Thus, we fix $\delta>0$ . If $x\stackrel {T}{\sim }x'$ and $y\stackrel {T}{\sim }y'$ , then there are $i,j,m,n\ge 0$ such that $T^i(x)=T^j(x')$ and $T^m(y)=T^n(y')$ . Using Fact 4.7 repeatedly, we have

$$ \begin{align*} \bar{f}_\delta(x,y)=\bar{f}_\delta(T^i(x),y)=\bar{f}_\delta(T^j(x'),y)=\bar{f}_\delta(x',y), \end{align*} $$

and similarly, $\bar {f}_\delta (x',y)=\bar {f}_\delta (x',y')$ .

By Fact 4.15 the Feldman–Katok pseudometric depends on the separation between forward orbits of given points, rather than on the points alone.

4.3 Comparison with the Besicovitch pseudometric

We note that our results about the Feldman–Katok pseudometric generalize those known for Besicovitch pseudometric (see [Reference Babel, Can, Kwietniak and Oprocha1, Reference Buldağ, Jacelon and Kwietniak11, Reference Kwietniak, Łącka and Oprocha36] for more details). In particular, any sequence of generic points converging in the Besicovitch pseudometric provides an example of an $\overline {fk}$ -convergent sequence of measures.

Proof. Define

$$ \begin{align*} \bar{d}_{n,\delta}(\underline{x},\underline{z})=\frac{1}{n}|\{0\le j< n : \rho(x_j, z_j)\ge\delta\}|. \end{align*} $$

If $D_{\mathrm {B}}'(\underline {x}, \underline {z})<\delta $ for some $\delta>0$ then for all n large enough $\bar {d}_{n,\delta }(\underline {x}, \underline {z})<\delta $ . It follows that there exists an $(n, \delta )$ -match of $\underline {x}$ with $\underline {z}$ . Therefore, $\overline {fk}(\underline {x}, \underline {z})\leq D_{\mathrm {B}}'(\underline {x}, \underline {z})$ .

5.1 ‘Completeness’ of $\overline {fk}$ -convergence

Since $\overline {fk}$ is a pseudometric on $X^{\infty} $ , given $T\colon X\to X$ , such notions as $\overline {fk}$ -Cauchy sequence or $\overline {fk}$ -limit have obvious meaning. The first important feature of the $\overline {fk}$ -convergence is that it has enough ‘completeness’ for our purposes: an $\overline {fk}$ -Cauchy sequence of orbits defines a quasi-orbit which is its $\overline {fk}$ -limit.

Repeating the proof of [Reference Blanchard, Formenti and Kůrka4, Proposition 2], we get the following fact.

In fact, the proof of Lemma 5.3 yields that there are $0=m_0<m_1<m_2<\cdots $ and a subsequence $(x^{(n_j)})_{j=1}^{\infty} $ of $(x^{(n)})_{n=1}^{\infty} $ such that $\bar {d}(\{m_j:j\ge 0\})=0$ and $\underline {z}$ is given by $z_n=T^n (x^{(n_j)})$ for $m_{j-1}\le n <m_{j}$ . We do not know whether under the assumptions of Lemma 5.3 there is a point whose orbit is an $\overline {fk}$ -limit of the $\overline {fk}$ -Cauchy sequence. We can prove this only by imposing an extra assumption on $T\colon X\to X$ . It turns out that the asymptotic average shadowing property is a sufficient requirement. Let us recall its definition.

Definition 5.4. A sequence $\underline {z}=(z_j)_{j=0}^{\infty} \in X^{\infty }$ is an asymptotic average pseudo-orbit for $T\colon X\to X$ if

$$ \begin{align*} D_{\mathrm{B}}(T(\underline{z}),\underline{z})=\lim\limits_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1} \rho(T(z_j),z_{j+1})=0. \end{align*} $$

We say that $\underline {z}=(z_j)_{j=0}^{\infty} \in X^{\infty }$ is asymptotically shadowed on average by $x\in X$ if

(7) $$ \begin{align} D_{\mathrm{B}}(\underline{x}_T,\underline{z})=\lim\limits_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\rho(T^j(x),z_j)=0. \end{align} $$

A system $(X,T)$ has the asymptotic average shadowing property if every asymptotic average pseudo-orbit of T is asymptotically shadowed on average by some point.

Note that if $\underline {x}_T$ and $\underline {z}$ satisfy (7), then $D_{\mathrm {B}}'(\underline {x}_T,\underline {z})=\overline {fk}(\underline {x}_T,\underline {z})=0$ as well (see (6) and Lemma 4.16). The asymptotic average shadowing property was introduced by Gu [Reference Rongbao41] and is implied by most versions of the specification property considered in the literature (see [Reference Can and Trilles14, Reference Kulczycki, Kwietniak and Oprocha33, Reference Kwietniak, Łącka and Oprocha35, Reference Kwietniak, Łącka and Oprocha36]).

Proof. Fix an $\overline {fk}$ -Cauchy sequence $(x^{(n)})_{n=1}^{\infty} \subset X$ . Note that the quasi-orbit $\underline {z}\in X^{\infty }$ provided by Lemma 5.3 is an asymptotic average pseudo-orbit and satisfies $\overline {fk}(\underline {x}_T,\underline {z})=0$ . Pick an $x\in X$ that asymptotically average-shadows $\underline {z}$ . Then

$$ \begin{align*} \overline{fk}(x,x^{(n)})\leq \overline{fk}(\underline{x}_T,\underline{z})+\overline{fk}(\underline{z},\underline{x}^{(n)}_T)\to 0\quad\text{as }n\to\infty. \end{align*} $$

Therefore, x is an $\overline {fk}$ -limit of $(x^{(n)})_{n=1}^{\infty} $ in X.

It turns out that the set of generic quasi-orbits is $\overline {fk}$ -closed. More is true: $V({\underline {x}})$ depends $\overline {fk}$ -continuously on $\underline {x}$ when we consider $V({\underline {x}})$ as a point in the space of non-empty closed subsets of ${\mathcal {M}}(X)$ endowed with the Hausdorff metric $D_{\mathrm {H}}$ induced by the Prokhorov metric $D_{\mathrm {P}}$ (the latter space is called the hyperspace of ${\mathcal {M}}(X)$ ).

Although the proof of the following fact is short, its importance justifies calling it a theorem.

An immediate consequence of Lemma 5.3 and Theorem 5.7 is the following corollary.

Corollary 5.8 allows us to justify the correctness of the next definition.

7.1 Auxiliary terminology and results

For $k\in \mathbb {N}$ , $\underline {x}=(x_j)_{j=0}^{\infty} \in X^{\infty }$ , and $\varphi \in \mathcal {C}(X)$ let $A_k(\varphi ,\underline {x})$ denote the Birkhoff average along $x_0,x_1,\ldots ,x_{k-1}$ , that is,

$$ \begin{align*} A_k(\varphi,\underline{x})=\frac{1}{k}\sum_{j=0}^{k-1}\varphi(x_j)=\int_X \varphi\,d{\mathfrak{m}}(\underline{x},k). \end{align*} $$

For $x\in X$ we write $A_k(\varphi ,x)$ for the Birkhoff average along an orbit segment of length k, that is, $A_k(\varphi ,x)=A_k(\varphi ,\underline {x}_T)$ .

Recall that a sequence $\underline {x}\in X^{\infty }$ (respectively, a point $x\in X$ ) is generic for some measure $\mu $ if and only if for every $\varphi \in \mathcal C(X)$ the sequence $A_k(\varphi ,\underline {x})$ (respectively, $A_k(\varphi ,x)$ ) converges as $k\to \infty $ . We denote the corresponding limit by $\varphi ^*(\underline {x})$ (respectively, $\varphi ^*(x)$ ). It is easy to see that for any $\ell \in \mathbb {N}$ , we have $\varphi ^*(\underline {x})=\varphi ^*(\sigma ^\ell (\underline {x}))$ (respectively, $\varphi ^*(x) =\varphi ^*(T^\ell (x))$ ). For a generic sequence $\underline {x}$ we put

$$ \begin{align*} A^*_k(\varphi,x_\ell)=|A_k(\varphi,(x_j)_{j=\ell}^{\infty})-\varphi^*((x_j)_{j=\ell}^{\infty})|=|A_k(\varphi,\sigma^\ell(\underline{x}))-\varphi^*(\underline{x})|. \end{align*} $$

Furthermore, for a generic point x and $\ell \in \mathbb {N}$ , we define

$$ \begin{align*} A^*_k(\varphi,T^\ell(x))=|A_k(\varphi,T^\ell(x))-\varphi^*(x)|. \end{align*} $$

Fix $\alpha>0$ , $k\in \mathbb {N}$ , and $\varphi \in \mathcal {C}(X)$ . We say that $\ell \ge 0$ is a starting point of an $(\alpha ,\varphi )$ -bad k-segment in a generic sequence $\underline {x}=(x_j)_{j=0}^{\infty} $ if $A_k^*(\varphi ,\sigma ^\ell (\underline {x}))>\alpha $ .

The following characterization of ergodic sequences slightly generalizes the one presented by Oxtoby in [Reference Oxtoby40, §4]. It states that a generic sequence $\underline {x}=(x_j)_{j=0}^{\infty} $ generates an ergodic measure if for every $\varphi \in \mathcal {C}(X)$ and $\alpha>0$ the upper density of the set of integers $\ell $ being starting points for an $(\alpha ,\varphi )$ -bad k-segments converges to zero as k goes to $\infty $ . Replacing $\underline {x}$ by $\underline {x}_T$ , we obtain a criterion for ergodicity of a measure given by a generic point due to Oxtoby. We omit the proof as it follows the same lines as in [Reference Oxtoby40].

Theorem 7.1. (Oxtoby’s criterion)

Let a quasi-orbit $\underline {z}=(z_j)_{j=0}^{\infty} \in X^{\infty }$ be generic for some $\mu \in {\mathcal {M}}_T(X)$ . Then $\mu $ is ergodic if and only if for every $\varphi \in \mathcal {C}(X)$ and $\alpha>0$ we have

$$ \begin{align*}\bar d(\{\ell\ge 0\,:\,|A_k(\varphi,\sigma^\ell(\underline{z}))-\varphi^*(\underline{z})|>\alpha\})\to 0\quad\text{as }k\to\infty.\end{align*} $$

Let $n\in \mathbb {N}$ , $x\in X$ and $\underline {z}\in X^{\infty }$ . For an $(n,\delta )$ -match $\pi $ of $\underline {x}_T$ with $\underline {z}$ and $k\le n$ we define for every $\ell \in \mathcal {D}(\pi )\cap \{0,1,\ldots ,n-k\}$ a set

$$ \begin{align*} \mathcal{D}^{\prime}_\ell= \{i\in\mathcal{D}(\pi): \ell\le i < \ell+k \text{ and }\pi(\ell)\le \pi(i) < \pi(\ell)+k \} \end{align*} $$

and the $(k,\delta )$ -match $\pi ^{\prime }_\ell \colon \mathcal {D}^{\prime }_\ell \to \pi (\mathcal {D}^{\prime }_\ell )$ induced by $\pi $ at $\ell $ setting $\pi ^{\prime }_\ell (i)=\pi (i)$ for ${i\in \mathcal {D}(\pi ^{\prime }_\ell )}$ . It is easy to see that $\pi ^{\prime }_\ell $ is indeed an $(k,\delta )$ -match of $T^\ell (x)$ with $z_{\pi (\ell )}$ satisfying $\mathcal {D}(\pi ^{\prime }_\ell )=\mathcal {D}^{\prime }_\ell $ and $\mathcal {R}(\pi ^{\prime }_\ell )=\pi (\mathcal {D}^{\prime }_\ell )$ .

We also need the following technical lemma, which asserts that if an orbit $\underline {x}_T$ and a quasi-orbit $\underline {z}$ are sufficiently $\overline {fk}$ -close, then for any $\varphi \in \mathcal C(X)$ and $k\in \mathbb {N}$ one can find a match $\pi $ which allows one to find a match $\pi $ of $\underline {x}_T$ and $\underline {z}$ so that the averages of $\varphi $ over k segments in $\underline {x}_T$ and $\underline {z}$ are also small for most of k-segments.

Lemma 7.2. Fix $\varphi \in \mathcal C(X)$ and $\varepsilon>0$ . Let $\delta>0$ be such that $y,y'\in X$ and ${\rho (y,y')<\delta }$ imply $|\varphi (y)-\varphi (y')|<\varepsilon $ . If $\underline {z}=(z_j)_{j=0}^{\infty} \in X^{\infty }$ is a quasi-orbit and $x\in X$ satisfies $\overline {fk}(\underline {x}_T,\underline {z})<\delta $ , then for every $k\in \mathbb {N}$ there exists $N\in \mathbb {N}$ such that for every $n\geq N$ there are an $(n,\delta )$ -match $\pi $ of $\underline {x}_T$ with $\underline {z}$ and a set $A\subset \mathcal D(\pi )$ with $|A|>n(1-2\sqrt {\delta }-2\delta )-k$ satisfying

$$ \begin{align*} |A_k(\varphi, T^\ell(x))-A_k(\varphi,\sigma^{\pi(\ell)}(\underline{z}))|\leq \varepsilon+4\sqrt\delta||\varphi||_{\infty}\quad\text{for every }\ell\in A. \end{align*} $$

Proof. Fix $k\in \mathbb {N}$ , $\varphi \in \mathcal C(X)$ and $\varepsilon>0$ . Choose $N\in \mathbb {N}$ such that for every $n\geq N$ there exists an $(n,\delta )$ -match $\pi $ of $\underline {x}_T$ with $\underline {z}$ satisfying $|\pi |>n(1-\delta )$ and

$$ \begin{align*}|\{0\leq j<n\,:T(z_{j+i})\neq z_{j+i+1}\text{ for some }0\leq i<k\}|<n\delta.\end{align*} $$

Fix $n\geq N$ . Define

$$ \begin{align*} A_Z & =\{0\leq j<n\,:T(z_{\pi(j)+i})\neq z_{\pi(j)+i+1}\text{ for some }0\leq i<k\},\\ A_R & =\{0\leq j<n-k\,:\,j\in\mathcal{D}(\pi)\text{ and }|\{0\leq i<k\,:\,\pi(j)+i\notin\mathcal{R}(\pi)\}|\geq\sqrt\delta k\},\\ A_D & =\{0\leq j<n-k\,:\,j\in\mathcal{D}(\pi)\text{ and }|\{0\leq i<k\,:\,j+i\notin\mathcal{D}(\pi)\}|\geq\sqrt\delta k\}.\end{align*} $$

Note that $|A_Z|\le n\delta $ . To estimate $|A_R|$ , define $\mathcal {R}^{c}(\pi )=\{0,1,\ldots ,n-1\}\setminus \mathcal {R}(\pi )$ , and note that

$$ \begin{align*} |\mathcal{R}^c(\pi)|\ge |\{m\in \mathcal{R}^c(\pi)\,:\,\pi(j)\le m <\pi(j)+k\text{ for some }j\in A_R\}|. \end{align*} $$

On the other hand, for each $m\in \mathcal {R}^c(\pi )$ the set $\{j\in A_R\,:\,\pi (j)\le m <\pi (j)+k\}$ has at most k elements; hence, each $j\in A_R$ implies that there are at least $\sqrt {\delta }$ members of $\mathcal {R}^c(\pi )$ . Thus, $|\mathcal {R}^c(\pi )|\ge |A_R| \sqrt {\delta }$ . This, together with $|\mathcal {R}^c(\pi )|<n\delta $ , gives us $|A_R|< n\sqrt \delta $ . An analogous reasoning leads to $|A_D|< n\sqrt \delta $ . Define

$$ \begin{align*} A=(\{0,\ldots, n-k-1\}\cap \mathcal D(\pi))\setminus(A_Z\cup A_D\cup A_R). \end{align*} $$

Then $|A|> n(1-2\sqrt \delta -2\delta )-k$ and for every $j\in A$ we have:

1. (i) the $(k,\delta )$ -match $\pi ^{\prime }_j$ induced by $\pi $ at j satisfies $|\pi ^{\prime }_j|>(1-2\sqrt {\delta })k$ ;

2. (ii) $A_k(\varphi ,z_{\pi (j)})=A_k(\varphi ,\sigma ^{\pi (j)}(\underline {z}))$ .

Therefore,

$$ \begin{align*} & |A_k(\varphi, T^j(x))-A_k(\varphi,\sigma^{\pi(j)}(\underline{z}))|=|A_k(\varphi, T^j(x))-A_k(\varphi,z_{\pi(j)})|\\ & \quad \leq\frac 1k\sum_{i\in \mathcal{D}(\pi^{\prime}_j)}|\varphi(T^{i}(x))-\varphi(z_{\pi(i)})|+\frac{1}{k}\sum_{i\notin \mathcal{D}(\pi^{\prime}_j)}|\varphi(T^{i}(x))|+\frac 1k\sum_{i\notin \mathcal{R}(\pi^{\prime}_j)}|\varphi(z_i)|\\ & \quad \leq \varepsilon+2\sqrt\delta||\varphi||_{\infty}+2\sqrt\delta||\varphi||_{\infty}=\varepsilon+4\sqrt\delta||\varphi||_{\infty} \end{align*} $$

and the lemma follows.

7.2 $\overline {fk}$ -limits of ergodic measures are ergodic

We prove the main theorem of this section. As the GIKN sequence of periodic orbits is $\overline {fk}$ -Cauchy by Theorem 6.2, we see that our Theorem 7.3 generalizes [Reference Gorodetskiĭ, Ilyashenko, Kleptsyn and Nalskiĭ29, Lemma 2]. Furthermore, since $\overline {fk}\le D_{\mathrm {B}}$ , this result also extends [Reference Kwietniak, Łącka and Oprocha36, Theorem 15].

Proof. By Fact 5.6 there exists a measure $\mu \in \mathcal {\mathcal {M}}_T(X)$ such that $\mu _p\to \mu $ as $p\to \infty $ in the weak $^*$ topology on ${\mathcal {M}}_T(X)$ . We will apply Oxtoby’s criterion (Theorem 7.1) to show that $\mu $ is ergodic. Let $\underline {z}=(z_j)_{j=0}^{\infty} $ be a quasi-orbit such that $\bar {f}(\underline {x}^{(p)}_T,\underline {z})\to 0$ as $p\to \infty $ provided by Lemma 5.3. Clearly, $\underline {z}$ is generic for $\mu $ . Fix $\varphi \in \mathcal {C}(X)$ and $\alpha>0$ . We need to show that for every $\eta>0$ and all sufficiently large k the set of js initiating an $(\alpha ,\varphi )$ -bad k-segment in $\underline {z}$ has upper density smaller than $\eta $ .

Note that for every $i,j,k,p\in \mathbb {N}$ we have

(8) $$ \begin{align} & |A_k(\varphi,\sigma^j(\underline{z})-\varphi^*(\underline{z}))|\leq |A_k(\varphi,\sigma^j(\underline{z}))-A_k(\varphi,T^i(x^{(p)}))|\nonumber\\ &\quad +|A_k(\varphi,T^i(x^{(p)}))-\varphi^*(x^{(p)})|+|\varphi^*(x^{(p)})-\varphi^*(\underline{z})|. \end{align} $$

By Theorem 5.7 and Fact 5.6 we can choose $P(\alpha )\in \mathbb {N}$ such that for every $p\geq P(\alpha )$ one has $|\varphi ^*(x^{(p)})-\varphi ^*(\underline {z})|\leq \alpha /3$ .

Let $\delta _0>0$ be such that $2\sqrt \delta _0+2\delta _0<\eta /4$ and $4\sqrt \delta _0||\varphi ||_{\infty }<\alpha /6$ . Pick $\delta <\delta _0$ such that for all $y,y'\in X$ with $\rho (y,y')<\delta $ one has $|\varphi (y)-\varphi (y')|<\alpha /6$ . Note that our choice of constants is motivated by Lemma 7.2. Choose $p\geq P(\alpha )$ such that $\bar {f}(\underline {z},x^{(p)}_T)<\delta $ . Pick $K>0$ such that for all $k\geq K$ the upper density of the set of is initiating an $(\alpha /3,\varphi )$ -bad k-segment in $x^{(p)}_T$ is smaller than $\eta /2$ . We are going to prove that each $k\ge K$ is ‘sufficiently large’ to imply that the upper density of the set of js initiating an $(\alpha ,\varphi )$ -bad k-segment in $\underline {z}$ is smaller than $\eta $ . To this end fix any $k\ge K$ . Let N be sufficiently large to guarantee that $k/N<\eta /4$ and for every $n\ge N$ we have

1. (i) there exists an $(n,\delta )$ -match $\pi $ of $\underline {z}$ with $x^{(p)}$ such that $|\pi |>(1-\delta )n$ ,

2. (ii) $|\{0\leq j<n\,:T(z_{j+i})\neq z_{j+i+1}\text { for some }0\leq i<k\}|<n\delta $ ,

3. (iii) $|\{0\le i<n\,:\, i$ initiates an $(\alpha /3,\varphi )$ -bad k-segment in $x^{(p)}_T\}|< n\eta /2$ .

We are going to show that for each $n\ge N$ the number of $0\le j<n$ such that

$$ \begin{align*} A^*_k(\varphi,\sigma^j(\underline{z}))=|A_k(\varphi,(z_s)_{s=j}^{\infty})-\varphi^*(\underline{z})|\leq\alpha \end{align*} $$

is larger than $(1-\eta )n$ . To this end let $\tilde \pi $ be any extension of $\pi $ to a bijection from $\{0,\ldots ,n-1\}$ onto $\{0,\ldots ,n-1\}$ . It follows from the proof of Lemma 7.2 that conditions (i) and (ii) guarantee that the number of integers $0\leq j<n$ for which

$$ \begin{align*} |A_k(\varphi,\sigma^j(\underline{z}))-A_k(\varphi,T^{\tilde\pi(j)}(x^{(p)}))|<\alpha/6+4\sqrt\delta_0||\varphi||_{\infty}<\alpha/3 \end{align*} $$

is larger than $n(1-2\sqrt \delta -2\delta )-k$ and hence larger than $n(1-\eta /2)$ . Some of these js may initiate an $(\alpha /3,\varphi )$ -bad k-segment in $x^{(p)}_T$ , but condition (iii) bounds from the above the number of such js by $n\eta /2$ . Therefore, setting $i=\tilde \pi (j)$ in (8), we see that the number of integers $0\leq j<n$ for which all summands on the right-hand side of (8) are bounded above by $\alpha /3$ is larger than $n(1-\eta )$ . Since this holds true for any $n\ge N$ we see that upper density of the set of all js initiating an $(\alpha ,\varphi )$ -bad k-segment in $\underline {z}$ is smaller than $\eta $ , hence the Oxtoby criterion yields that $\underline {z}$ is an ergodic sequence.

The ergodicity of GIKN measures is now a straightforward corollary; hence [Reference Bonatti, Díaz and Gorodetski8, Lemma 2.5] is recovered, except for the statement about the support (which can now be proved in the same way as in [Reference Bonatti, Díaz and Gorodetski8], since the proof of that part of [Reference Bonatti, Díaz and Gorodetski8, Lemma 2.5] is independent of the rest).

9.1 Katok’s criterion

Let $\mathbf {X}=(X,\mathscr X, \mu , T)$ be a measure-preserving system and let $\mathcal P=\{P_0,P_1,\ldots , P_{k-1}\}\in \mathbf {P}^k(X)$ . Following Katok [Reference Katok30, Definition 9.1], we say that the process $(\mathbf {X}, \mathcal P)$ is $(n,\varepsilon )$ -trivial if there exists a word $\omega \in \{0,1,\ldots , k-1\}^n$ such that $\mu _{{\mathcal {P}}}(B_{\varepsilon }[\omega ])\geq 1-\varepsilon $ , where $B_{\varepsilon }[\omega ]=\{\omega '\in \{0,1,\ldots , k-1\}^n\,:\,\bar {f}_{n}(\omega , \omega ')<\varepsilon \}$ . For every $n\ge 1$ , $\varepsilon>0$ and $\omega \in \{0,1\ldots ,k-1\}^n$ we clearly have

$$ \begin{align*} (1-\mu_{{\mathcal{P}}}(B_{\varepsilon}[\omega]))\varepsilon\le \int_X \bar{f}_n({\mathcal{P}}^n(x),\omega){\, d}\mu(x)\le (1-\mu_{{\mathcal{P}}}(B_{\varepsilon}[\omega]))+\varepsilon. \end{align*} $$

Therefore, (cf. [Reference Katok30, Lemma 9.1]) if $\omega \in \{0,1,\ldots ,k-1\}^n$ and $\beta>0$ are such that

$$ \begin{align*} \int_X \bar{f}_n({{\mathcal{P}}}^n(x),\omega)\,\textit{d}\mu(x)=\int_{\Omega_k} \bar{f}_n(u_0u_1\cdots u_{n-1},\omega)\,\textit{d}\mu_{\mathcal{P}}(u)<\beta, \end{align*} $$

then the process $(\mathbf {X}, \mathcal P)$ is $(n, \sqrt \beta )$ -trivial. Conversely, if the process $(\mathbf {X}, \mathcal P)$ is $(n,\varepsilon /2)$ -trivial, then there is $\omega ^{(n)} \in {\mathcal {P}}^n$ such that

$$ \begin{align*}\int_X \bar{f}_n({\mathcal{P}}(x), \omega^{(n)}){\, d}\mu(x)=\int_{\Omega_k} \bar{f}_n(u_0u_1\cdots u_{n-1},\omega^{(n)})\,\text{d}\mu_{\mathcal{P}}(u)<\varepsilon.\end{align*} $$

A process $(\mathbf {X},{\mathcal {P}})$ is M-trivial [Reference Katok30, Definition 9.2] if for any $\varepsilon>0$ there exists $N=N(\varepsilon )$ such that for every $n\geq N$ the process $(\mathbf {X},\mathcal P)$ is $(n,\varepsilon )$ -trivial.

9.2 Proof of Theorem 9.1

In the proof of our main theorem, we will need the following fact.

Lemma 9.3. Let $\mu \in {{\mathcal {M}}_T^e}(X)$ and $\varepsilon>0$ . If ${\mathcal {P}},\mathcal {R}\in \mathbf {P}^k(X)$ are such that $d^\mu _1({\mathcal {P}},\mathcal {R})<\varepsilon /3$ , then there is $N\in \mathbb {N}$ such that for every $n\ge N$ we have

$$ \begin{align*} \int_X \bar{f}_n({\mathcal{P}}^n(x),\mathcal{R}^n(x)){\, d}\mu(x)< \varepsilon. \end{align*} $$

Proof. Let ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{k-1}\}$ , $\mathcal {R}=\{R_0,R_1,\ldots ,R_{k-1}\}$ and $\Delta =(P_0 \div R_0)\cup (P_1 \div R_1)\cup \cdots \cup (P_{k-1} \div R_{k-1})$ . Observe that the ergodicity of $\mu $ implies that for $\mu $ -almost every $x\in X$ we have

$$ \begin{align*} \bar{d}_n({\mathcal{P}}^n(x),\mathcal{R}^n(x)) & =\frac{1}{n}\sum_{j=0}^{n-1}\chi_\Delta(T^j(x))\stackrel{(n\to\infty)}{\longrightarrow} \mu(\Delta)\\ & = \mu(\{x\in X:{\mathcal{P}}(x)\neq\mathcal{Q}(x)\})=d^\mu_1({\mathcal{P}},\mathcal{Q}). \end{align*} $$

It follows from the Egorov theorem that there are a measurable set $G\subset X$ and $N\in \mathbb {N}$ such that $\mu (G)>1-\varepsilon /3$ and for all $n\geq N$ and $x\in G$ one has $\bar {d}_n({\mathcal {P}}^n(x),\mathcal {R}^n(x))<d^\mu _1({\mathcal {P}},\mathcal {R})+\varepsilon /3$ . Notice that $\bar {f}_n({\mathcal {P}}^n(x),\mathcal {R}^n(x))\le \bar {d}_n({\mathcal {P}}^n(x),\mathcal {R}^n(x))$ for all $n\in \mathbb {N}$ . Therefore, if $d^\mu _1({\mathcal {P}},\mathcal {R})<\varepsilon /3$ and $n\ge N$ , then

$$ \begin{align*} \begin{aligned} \int_X \bar{f}_n({\mathcal{P}}^n(x),\mathcal{R}^n(x)){\, d}\mu(x) &\le \int_X \bar{d}_n({\mathcal{P}}^n(x),\mathcal{R}^n(x)){\, d}\mu(x)\\ &\le \int_G \bar{d}_n({\mathcal{P}}^n(x),\mathcal{R}^n(x)){\, d}\mu(x) +\mu(X\setminus G)\\ &\le d^\mu_1({\mathcal{P}},\mathcal{R})+\varepsilon/3+\varepsilon/3 \le\varepsilon. \end{aligned}\\[-39pt] \end{align*} $$

The rest of this section is devoted to the proof of Theorem 9.1.

Proof of Theorem 9.1

Let $\mu =\mu ^{(0)}$ be an $\overline {fk}$ -limit of a sequence of loosely Kronecker measures $\mu ^{(1)},\mu ^{(2)},\ldots .$ Then there are a sequence $(x^{(n)})_{n=1}^{\infty} \subseteq X$ and a generic quasi-orbit $\underline {x}$ for $\mu ^{(0)}$ such that $\overline {fk}(\underline {x},\underline {x}^{(n)}_T)\to 0$ as $n\to \infty $ and $x^{(n)}$ is a generic point for $\mu ^{(n)}$ for $n\in \mathbb {N}$ . Then $\mu ^{(0)}$ is ergodic. If $\mu ^{(0)}$ is a periodic measure, then there is nothing to show. Assume that $\mu ^{(0)}$ is aperiodic.

To apply Katok’s criterion (Theorem 9.2) we need to show that for every finite partition $\mathcal P$ of X the process $(\mathbf {X},\mathcal P)$ , where $\mathbf {X}=(X,X_{\mathscr {B}},\mu ^{(0)},T)$ , is M-trivial. To this end we choose a partition ${\mathcal {P}}\in \mathbf {P}^k(X)$ and fix $\varepsilon>0$ .

We use Lemma 2.3 to find $0<\alpha <\varepsilon /6$ such that $\bar {f}(\underline {\omega },\underline {\omega }')<\alpha $ for some $\underline {\omega }\in \operatorname {\mathrm {Gen}}(\xi )$ , $\underline {\omega }'\in \operatorname {\mathrm {Gen}}(\xi ')$ , where $\xi $ and $\xi '$ being shift-invariant ergodic measures on $\Omega _{k+1}$ implies that $\bar {f}(\xi ,\xi ')<\varepsilon /3$ .

We take $\delta <\min \{\alpha /2,\varepsilon /9\}$ . We apply Lemma 8.2 to $(\mu ^{(n)})_{n=0}^{\infty} $ to find $\gamma =\gamma (\mu =\mu ^{(0)},\delta ,{\mathcal {P}})$ and a partition $\mathcal {R}\in \mathbf {P}^{k+1}(X)$ with $d_1^{\mu ^{(0)}}({\mathcal {P}},\mathcal {R})<\delta $ and $\mu ^{(n)}(\partial \mathcal R)=0$ for ${n=0,1,\ldots .}$ Let N be such that $\overline {fk}(\underline {x},x^{(N)}_T)<\min \{\gamma ,\alpha /2\}$ . It follows from Lemma 8.2 that

(9) $$ \begin{align} \bar{f}(\mathcal{R}(\underline{x}),\mathcal{R}(\underline{x}^{(N)}_T))<\alpha<\varepsilon/6. \end{align} $$

By Lemma 2.2, the point $\mathcal {R}(\underline {x}^{(N)}_T)$ is generic for $\mathcal {R}$ -representation measure of $\mu ^{(N)}$ denoted by $\mathcal {R}_*\mu ^{(N)}=\mu ^{(N)}_{\mathcal {R}}$ and $\mathcal {R}(\underline {x})$ is generic for the $\mathcal {R}$ -representation $\mu ^{(0)}_{\mathcal {R}}=\mathcal {R}_*\mu ^{(0)}$ of $\mu ^{(0)}$ . Inequality (9) and our choice of $\alpha $ imply that $\bar {f}(\mu ^{(0)}_{\mathcal {R}},\mu _{\mathcal {R}}^{(N)})<\varepsilon /3$ . Hence, by the definition of the $\bar {f}$ -metric, we find $M'=M'(\varepsilon )>0$ satisfying that for every $m\ge M'$ there is a coupling $\unicode{x3bb} _{m}$ of $\mathcal {R}_*\mu |_{\mathcal {R}^m}$ and $\mathcal {R}_*\mu ^{(N)}|_{\mathcal {R}^m}$ for which we have

(10)

Using that $\mu ^{(N)}$ is loosely Kronecker or periodic, we find $M"=M"(\varepsilon )$ such that for every $m\ge M"$ there is $\omega ^{(m)}\in \mathcal {R}^m$ satisfying

(11) $$ \begin{align} \int_X \bar{f}_m(\mathcal{R}^m(x),\omega^{(m)}){\, d}\mu^{(N)}(x)<\varepsilon/3. \end{align} $$

With the above notation, let $m\ge \max \{M',M"\}$ . Using (10) and (11), we get (below, ${\mathcal {R}^{2m}=\mathcal {R}^m\times \mathcal {R}^m}$ )

(12)

By Lemma 9.3, since $\mathcal {R}$ is a partition satisfying $d^{\mu ^{(0)}}_1({\mathcal {P}},\mathcal {R})<\delta <\varepsilon /9$ , there exists $M\in \mathbb {N}$ such that for every $m\ge M$ we have

(13) $$ \begin{align} \int_X \bar{f}_m({\mathcal{P}}^m(x),\mathcal{R}^m(x)){\, d}\mu^{(0)}(x)<\varepsilon/3. \end{align} $$

Combining (11) with (12), we see that if $m\ge N=\max \{M,M',M"\}$ , then

$$ \begin{align*} \int_X \bar{f}_m({\mathcal{P}}^m(x),\omega^{(m)}){\, d}\mu^{(0)}(x)<\varepsilon. \end{align*} $$

Since $\varepsilon>0$ was arbitrary, we have proved that $(\mathbf {X},{\mathcal {P}})$ is M-trivial. Since ${\mathcal {P}}$ was also arbitrary and $\mu ^{(0)}$ is aperiodic we conclude, using Theorem 9.2, that $\mu ^{(0)}$ is loosely Kronecker (standard).