On the prevalence of the periodicity of maximizing measures

TL;DR

This paper demonstrates that for a specific dynamical system, the property that maximizing measures are supported on periodic orbits is common among Lipschitz functions, supporting conjectures in ergodic optimization.

Contribution

It proves the prevalence of the periodicity property for maximizing measures in Lipschitz functions on the full shift, strengthening previous results and confirming experimental predictions.

Findings

Property $ ext{ extbackslash mathscr ext{P}}_T$ is prevalent in Lipschitz functions.

Supports the Hunt--Ott conjectures in ergodic optimization.

Strengthens prior results by extending the class of functions and metrics.

Abstract

For a continuous map $T: X\rightarrow X$ on a compact metric space $(X,d)$, we say that a function $f: X \rightarrow \mathbb{R}$ has the property $\mathscr{P}_T$ if its time averages along forward orbits of $T$ are maximized at a periodic orbit. In this paper, we prove that for the one-side full shift of two symbols, the property $\mathscr{P}_T$ is prevalent (in the sense of Hunt--Sauer--Yorke) in spaces of Lipschitz functions with respect to metrics with mildly fast decaying rate on the diameters of cylinder sets. This result is a strengthening of \cite[Theorem~A]{BZ16}, confirms the prediction mentioned in the ICM proceeding contribution of J. Bochi (\cite[Seciton 1]{Boc18}) suggested by experimental evidence, and is another step towards the Hunt--Ott conjectures in the area of ergodic optimization.

Equations

$$P_{\mu }(T,f)=h_{\mu }(T)+\int f\mathrm{d}\mu ,$$

$$\beta (f):=\underset{\mu \in \mathcal{M}(X,T)}{\sup }\int f\mathrm{d}\mu =\underset{\mu \in \mathcal{M}(X,T)}{\max }\int f\mathrm{d}\mu .$$

$$\mathcal{P}(T)$$

$$\mathcal{PU}(T)$$

$$\sup \{|f(x)-f(y)|:x,y\in X,d(x,y)\le r\}=O(r)\text{ as }r\to 0.$$

$$\sup \{|f(x)-f(y)|:x,y\in X,d(x,y)\le r\}=o(r)\text{ as }r\to 0.$$

$$\Vert f{\Vert }_{\mathrm{Lip}}:=\Vert f{\Vert }_{\infty }+{\mathrm{LIP}}_d(f),$$

$${\mathrm{LIP}}_d(f)=\sup \{|f(x)-f(y)|/d(x,y):x,y\in X,x\ne y\}.$$

$$\sigma (\{w_i{\}}_{i\in \mathbb{N}})=\{w_{i+1}{\}}_{i\in \mathbb{N}}\text{for }\{w_i{\}}_{i\in \mathbb{N}}\in {\Sigma }^{+}.$$

$$d_{\mathbf{a}}(\mathbf{w},\mathbf{z}):=a_m$$

$$\frac{a_{n+1}}{a_n}=O({\theta }^n)\text{ as }n\to +\infty .$$

$$\frac{a_{n+1}}{a_{n}}=O\bigl{(}2^{-2^{n+2}}\bigr{)}\qquad\text{ as }n\to+\infty.$$

$$\sigma (\{w_i{\}}_{i\in \mathbb{N}})=\{w_{i+1}{\}}_{i\in \mathbb{N}}\text{for }\{w_i{\}}_{i\in \mathbb{N}}\in {\Sigma }^{+}.$$

$$\mathbf{x}†\mathbf{y}=\mathbf{y}†\mathbf{x},\mathbf{x}†\mathbf{z}\ge \min \{\mathbf{x}†\mathbf{y},\mathbf{y}†\mathbf{z}\}.$$

$$\text{mbox}{var}_n(f):=\underset{\mathbf{x}†\mathbf{y}\ge n}{\sup }|f(\mathbf{x})-f(\mathbf{y})|.$$

$$h_{\mathbf{w}}:=\frac{\text{mathbbold}{1}_{[\mathbf{w}0]}-\text{mathbbold}{1}_{[\mathbf{w}1]}}{2}.$$

$$f=c_{\varnothing }(f)+\sum _{\mathbf{w}\in {\Sigma }_{\ast }}{c}_{\mathbf{w}}(f)h_{\mathbf{w}}\text{(equality in L2(β)),}$$

$$c_{\varnothing }(f):=\int f\mathrm{d}\beta ,c_{\mathbf{w}}(f):=2^{|\mathbf{w}|+2}\int f{h}_{\mathbf{w}}\mathrm{d}\beta ,$$

$$A_{n}f=c_{\varnothing }(f)+\sum _{|\mathbf{w}|<n}{c}_{\mathbf{w}}(f)h_{\mathbf{w}};$$

$$A_{n}f=\sum _{\mathbf{w}\in {\Sigma }_n}\left(2^n{\int }_{[\mathbf{w}]}f\mathrm{d}\beta \right)\text{mathbbold}{1}_{[\mathbf{w}]}.$$

$$\Vert f-A_{n}f{\Vert }_{\infty }\le \text{mbox}{var}_n(f),$$

$$f=c_{\varnothing }(f)+\sum _{n=0}^{+\infty }\sum _{\mathbf{w}\in {\Sigma }_n}{c}_{\mathbf{w}}(f)h_{\mathbf{w}},$$

$$c_{\mathbf{w}}(f)\le {\mathrm{var}}_n(f).$$

$${\mathrm{LIP}}_{d_{\mathbf{a}}}(f)=\underset{n\in \mathbb{N}}{\sup }\frac{{\mathrm{var}}_n(f)}{a_n}.$$

$${\overline{b}}_n=o(a_n)\text{as }n\to +\infty ,$$

$${\overline{b}}_n:=\underset{\mathbf{w}\in {\Sigma }_n}{\max }\{b_{\mathbf{w}}\}\text{for each }n\in {\mathbb{N}}_0.$$

$${\mathcal{H}}_{\mathbf{b}}:=\{f_{\mathbf{e}}:\mathbf{e}=\{e_{\mathbf{w}}{\}}_{\mathbf{w}\in {\Sigma }_{\ast }}\text{ with }|e_{\mathbf{w}}|\le b_{\mathbf{w}}\text{ for each }\mathbf{w}\in {\Sigma }_{\ast }\}$$

$${\mathbb{P}}_{\mathbf{b}}(\{g\in {\mathcal{H}}_{\mathbf{b}}:f_0+g\in {\mathrm{Lock}}_{\sigma }({\Sigma }^{+},d_{\mathbf{a}})\})=1,$$

$$W_n^f(\underset{\mathbf{w}}{\underbrace{x_1{x}_2\dots x_n}})=\frac{1}{2}\sum _{i=0}^{n-1}(-1{)}^{x_{i+1}}{c}_{x_1{x}_2\dots x_i}(f).$$

$$M_n^f(\mathcal{C}):=\frac{1}{\mathrm{len}(\mathcal{C})}\sum _{\mathbf{w}\in A(\mathcal{C})}{W}_n^f(\mathbf{w})$$

$$W_n^f(\mathbf{w})$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Full shift

4. 2.2 Haar series

5. 2.3 Gauges

6. 3 A concrete formulation of Theorem 1.1

7. 4 A gap criterion for the locking property

8. 4.1 Graph theory

9. 4.2 Gap criterion

10. 5 Random perturbations on the edge-weight

11. 6 Proof of Theorem 3.1

12. 7 Proofs of Theorem 1.2 and Theorem 1.3

1. Introduction

Mathematicians have hoped for an analog of the notions of “Lebesgue almost every” and “Lebesgue measure zero” in infinite-dimensional spaces such as various Banach spaces of functions studied in dynamics. Such a desire was recorded in the contribution of A. N. Kolmogorov to the 1954 International Congress of Mathematicians [Ko54].

A natural notion to fulfill this quest called prevalence was introduced by B. R. Hunt, T. Sauer, and J. A. Yorke [HSY92]. This notion has since proven to play a central role in understanding generic behaviors in dynamics from a probabilistic (measure-theoretic) perspective. For more history, related notions, and applications in analysis, dynamics, economics, etc., we refer the readers to the beautiful surveys by B. R. Hunt and V. Yu. Kaloshin [HK10] and by W. Ott and J. A. Yorke [OY05].

Following [HK10], we recall the notion of prevalence as follows.

Let $V$ be a topological vector space over $\mathbb{K}\in\{\mathbb{R},\,\mathbb{C}\}$ equipped with a complete metric. A Borel measure $\mu$ on $V$ is called transverse to a Borel set $B\subseteq V$ if $\mu(\{x+v:x\in B\})=0$ for every $v\in V$. We say that a Borel subset $B$ of $V$ is shy if there exists a compactly supported Borel probability measure transverse to $B$. A subset $S$ of $V$ is called prevalent (resp. shy) if $V\setminus S$ (resp. $S$) is contained in a shy Borel subset of $V$.

Prevalence has the following properties (see for example, [HK10, Section 2.1]): it is preserved under translation and countable intersections, it implies density, and it coincides with the notion of having full Lebesgue measure when the ambient space is a finite-dimensional Euclidean space.

The aim of this paper is to investigate the prevalence of some ergodic properties in the study of dynamical systems.

We review some notions in ergodic theory. Let $T\colon X\rightarrow X$ be a continuous transformation. Let $f\colon X\rightarrow\mathbb{R}$ be a continuous function, called a potential. Recall the measure-theoretic pressure

[TABLE]

where $h_{\mu}(T)$ is the measure-theoretic entropy. In the language of statistical mechanics, the measure-theoretic entropy and the integral of the potential represent the kinetic and potential energy, respectively, while the measure-theoretic pressure represents the free energy, which is the sum of the kinetic energy and the potential energy. Under this interpretation, a measure of maximal entropy maximizes the kinetic energy (i.e., $h_{\mu}(T)$), and an equilibrium state maximizes the free energy (i.e., $P_{\mu}(T,f)$). Vast literature has been devoted to the studies of these measures for over half a century now. It is thereby only natural to consider measures of maximal potential energy.

We denote the maximal potential energy by

[TABLE]

The last identity follows from the weak*-compactness of the set $\mathcal{M}(X,T)$ of $T$-invariant Borel probability measures on $X$. The quantity $\beta(f)$ can also be expressed in terms of the maximal time average of $f$ (see for example, [Je06, Proposition 2.1]). We call a measure $\mu\in\mathcal{M}(X,T)$ that maximizes the potential energy $\int\!f\,\mathrm{d}\mu$ a measure of maximal potential energy with respect to $f$ or a $f$-potential-energy-maximizing measure (or a $f$-maximizing measure).

If one of the $f$-maximizing measures is supported on a periodic orbit of $T$, then we say that $f$ has property $\mathscr{P}_{T}$ and call such a measure periodic. We define the following subsets of the set $C(X)$ of real-valued continuous functions:

[TABLE]

The set of functions with property $\mathscr{P}_{T}$ have been investigated in various Banach spaces of continuous functions for many dynamical systems, see for example, [Bou00, Bou01, CLT01, QS12, Co16, BZ16, HLMXZ19, LZ23]. The study of typical properties of maximizing measures for Lipschitz and Hölder functions has been particularly fruitful.

Assume that $(X,d)$ is a compact metric space with infinite cardinality. Recall that the space $\operatorname{Lip}(X,d)$ of real-valued Lipschitz functions consists of functions $f\colon X\rightarrow\mathbb{R}$ with

[TABLE]

Similarly, a function $f\colon X\rightarrow\mathbb{R}$ is a little Lipschitz function or locally flat Lipschitz function if

[TABLE]

The space of real-valued little Lipschitz functions on $(X,d)$ is denoted by $\operatorname{lip}(X,d)$ and called the little Lipschitz space. Clearly $\operatorname{lip}(X,d)\subseteq\operatorname{Lip}(X,d)$. We equip both spaces with the the Lipschitz norm

[TABLE]

i.e., the sum of the sup-norm and the minimal Lipschitz constant $\operatorname{LIP}_{d}(f)$ of $f$ given by

[TABLE]

Then $\operatorname{Lip}(X,d)$ and $\operatorname{lip}(X,d)$ are Banach spaces. Moreover, $\operatorname{Lip}(X,d)$ and $\operatorname{lip}(X,d^{\alpha})$, $\alpha\in(0,1)$, are non-separable unless $X$ is finite or countably infinite, because they contain the discrete subset $\{d(x,\mathord{\cdot}):x\in X\}$. Here the snowflake $d^{\alpha}$ of $d$ given by $d^{\alpha}(x,y)\coloneqq d(x,y)^{\alpha}$ is also a metric for each $\alpha\in(0,1]$. The space $C^{0,\alpha}(X,d)$ of $\alpha$-Hölder continuous functions is precisely $\operatorname{Lip}(X,d^{\alpha})$. The space $c^{0,\alpha}(X,d)\coloneqq\operatorname{lip}(X,d^{\alpha})$ is called the space of little $\alpha$-Hölder functions by some authors.

The Lipschitz space $\operatorname{Lip}(X,d)$ is sometimes also called the big Lipschitz space. Little Lipschitz spaces, like big Lipschitz spaces, play central roles in the study of Lipschitz analysis and Lipschitz algebras in the analysis on metric spaces and functional analysis; see the beautiful monographs of N. Weaver [We18] and of Ş. Cobzaş, R. Miculescu, and A. Nicolae [CMN19], as well as references to the vast literature therein. Among other important properties, little Lipschitz spaces are double preduals of the big Lipschitz spaces under mild assumptions (see for example, [We18, Section 4.3]). We also refer the readers to [CMN19, Sections 8.3 to 8.5] for some connections between little Lipschitz spaces and mass transport problems, which is another source of ideas and tools for the investigations in ergodic optimization (c.f. [BZ15]).

We say that a function $f\in\operatorname{Lip}(X,d)$ (resp. $\operatorname{lip}(X,d)$) has the locking111This is translated from the term verrouillage used by T. Bousch in [Bou00, Secton 8]. property if it has a unique maximizing measure $\nu\in\mathcal{M}(X,T)$ that is supported on a periodic orbit of $T$ (in particular, $f$ has property $\mathscr{P}_{T}$) and moreover $\nu$ is the unique maximizing measure for every function in $\operatorname{Lip}(X,d)$ (resp. $\operatorname{lip}(X,d)$) sufficiently close to $f$ in the Lipschitz norm. The set of Lipschitz functions $f\in\operatorname{Lip}(X,d)$ (resp. $\operatorname{lip}(X,d)$) with the locking property is denoted by $\operatorname{Lock}_{T}(X,d)$ (resp. $\operatorname{lock}_{T}(X,d)$). The locking property clearly depends on the metric $d$.

The set of Lipschitz functions with the locking property is exactly the interior of the set of Lipschitz functions with property $\mathscr{P}_{T}$. One inclusion is obvious from the definition, and the other was shown by G. Yuan and B. R. Hunt [YH99, Remark 4.5] (see also [BZ15]). The same property holds for little Lipschitz functions, see [LZ23, Theorem 4.1]. In particular, the functions with the locking property is dense among the functions with property $\mathscr{P}_{T}$.

Consider the full shift $(\Sigma^{+},\sigma)$ on the set $\Sigma^{+}\coloneqq\{0,\,1\}^{\mathbb{N}}$ of binary sequences. Here the left-shift operator $\sigma\colon\Sigma^{+}\rightarrow\Sigma^{+}$ is given by

[TABLE]

For a strictly decreasing sequence $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ of positive numbers converging to zero, define a function $d_{\bm{a}}\colon\Sigma^{+}\times\Sigma^{+}\rightarrow\mathbb{R}$ by assigning, for each pair of distinct $\bm{w}=\{w_{n}\}_{n\in\mathbb{N}},\,\bm{z}=\{z_{n}\}_{n\in\mathbb{N}}\in\Sigma^{+}$,

[TABLE]

where $m$ is the smallest positive integer with the property that $w_{n}\neq z_{n}$, and by setting $d_{\bm{a}}(\bm{z},\bm{z})\coloneqq 0$ for each $\bm{z}\in\Sigma^{+}$. It is easy to see that $d_{\bm{a}}$ is a metric on $\Sigma^{+}$.

We consider the non-separable Banach spaces $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ and $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$ with respect to the metric $d_{\bm{a}}$. Observe that the faster the sequence ${\bm{a}}$ decays, the more regular the functions in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ are.

We are now ready to state our main theorem below.

Theorem 1.1** (Main Theorem).**

Let $(\Sigma^{+},\sigma)$ be the full shift on binary sequences. Let $\theta$ be a number in $(0,1/4)$ and $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ be a strictly decreasing sequence of positive numbers satisfying

[TABLE]

Then $\mathscr{P}\mathscr{U}(\sigma)\cap\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in the space $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ of real-valued Lipschitz functions, where $\mathscr{P}\mathscr{U}(\sigma)$ consists of continuous functions $f\colon\Sigma^{+}\rightarrow\mathbb{R}$ with the property that there is a unique $f$-maximizing measure $\mu$ and that $\mu$ is supported on a periodic orbit of $\sigma$. Moreover, $\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$.

In general, prevalence does not guarantee topological genericity, neither does the latter imply the former. However, due to the fact that $\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is known to be an open set in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$, topological genericity can be guaranteed by the density alone, which is always a direct consequence of prevalence. Therefore, Theorem 1.1 is, in our special setting, a strengthening of the topological result in [Co16, Theorem A].

As explained in [Boc18, Section 1], the type of results like Theorem 1.1 provide some confirmation on the experimental findings of B. R. Hunt and E. Ott published over two decades ago [HO96a, HO96b].

In the late 1990s, B. R. Hunt, E. Ott, and G. Yuan conjectured that for a typical chaotic system $T$, a typical smooth function $f$ admits a maximizing measure supported on a periodic orbit, where the typicality on the function $f$ was both in the sense of probability (in terms of the Lebesgue measure of the parameter(s) for the potential $f$ in a parameter space) ([HO96a, Conjecture] and [HO96b, Conjecture 2]) and in the sense of topology (topological genericity) ([YH99, Conjecture 1.1]). We recall that such a function $f$ has property $\mathscr{P}_{T}$. Moreover, B. R. Hunt and E. Ott observed based on numerical evidence that the functions $f$ without property $\mathscr{P}_{T}$ has fractal dimension zero. Despite the recent breakthroughs in the direction of topological genericity initiated by [Co16], the Hunt–Ott conjecture has been much less thoroughly investigated.

For a specific map $T(x)=2x$ on the circle, and a parameter family $\rho_{\tau}(x)=\cos 2\pi(x-\tau)$, Corollary 1 in [Bou00] shows that the maximizing measure is unique and that it is supported on a periodic orbit for every $\tau$ except on a set which has zero Lebesgue measure and zero Hausdorff dimension (see also [Je00]). Therefore, the Hunt–Ott conjecture is confirmed in this specific one-dimensional parameter family. The notion of full Lebesgue measure coincides in finite dimension with the notion of prevalence in Theorem 1.1.

In fact, Theorem 1.1 is a strengthened version of [BZ16, Theorem A] by J. Bochi and the third-named author, where they assume the evanescent condition on $\bm{a}$, i.e., $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ is a strictly decreasing sequence of positive numbers satisfying

[TABLE]

Theorem 1.1 guarantees the prevalence of continuous functions with the locking property in bigger subspaces of Hölder continuous functions (than the space of functions satisfying (1.4)), which confirms the prediction mentioned in [Boc18, Seciton 1] suggested by experimental evidence, and is another step towards the Hunt–Ott conjectures.

As a counterpart to Theorem 1.1, the following result for little Lipschitz functions holds.

Theorem 1.2**.**

Assume the hypotheses of Theorem 1.1. Then $\mathscr{P}\mathscr{U}(\sigma)\cap\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$. Moreover, $\operatorname{lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$.

We remark that by a theorem of the second-named author and the third-named author [LZ23], we know that $\operatorname{lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is an open and dense subset of $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$. We also remark that the prevalence of the subset of potentials with unique maximizing measure in the space of all continuous functions was established in [Mo21].

For another application, we recall that a measure $\mu\in\mathcal{M}(\Sigma^{+},\sigma)$ is a ground state for $\sigma$ and a potential $f$ if it is the limit of a sequence of equilibrium states $\mu_{t_{i}f}$ in the weak∗ topology for some sequence $\{t_{i}\}_{i\in\mathbb{N}}$ of real numbers that tends to infinity. Since $t=t_{i}$ can be considered as inverse temperature, we say that equilibrium state $\mu_{tf}$ converges at zero temperature if there exists a unique ground state.

Theorem 1.3** (Prevalence of the uniqueness of ground states).**

Assume the hypotheses of Theorem 1.1. Then the following statements hold:

1. (i)

The set of $f$ in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ that admits a unique ground state (i.e., the equilibrium state $\mu_{tf}$ converges at zero temperature) is a prevalent set in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. 2. (ii)

The set of $g$ in $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$ that admits a unique ground state is a prevalent set in $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$.

Remark 1.4**.**

Since $d_{\bm{a}}^{\alpha}=d_{\bm{a}^{\alpha}}$ where $\bm{a}^{\alpha}\coloneqq\{a_{n}^{\alpha}\}_{n\in\mathbb{N}_{0}}$, it is easy to see that we can replace the metric $d_{\bm{a}}$ by $d_{\bm{a}}^{\alpha}$ for $\alpha\in(-\log_{\theta}4,1]$ in Theorems 1.1 through 1.3 to extend these results to Hölder continuous functions with exponent sufficiently close to $1$.

Main ideas of the proof. The strategy of the proof of Theorem 1.1 relies on an approach of finite-dimensional approximations via Haar wavelets for ergodic optimization, which can be conveniently re-expressed as a maximum mean cycle problem on de Bruijn–Good directed graphs. Due to the hypothesis (1.3), the coefficients in the Haar series for a Lipschitz function (with respect to $d_{\bm{a}}$) decay fast in a controlled way. Our main goal is to show that with high probability, the maximizing measure for a function $f$ coincides with that for a step function obtained from truncating the Haar series for $f$ at some finite step in the finite-dimensional approximations. To this end, we adapt a strategy from [CCK15, Section 5], and make refined estimates to get a lower bound for the fluctuation on the (conditional) gap between the mean-weights of the two heaviest directed cycles on the sequence of de Bruijn–Good directed graphs. These estimates ensure that the hypothesis (1.3) is sufficient to guarantee the prevalence of the periodicity of the maximizing measure.

De Bruijn–Good directed graphs were independently introduced by N. G. de Bruijn [deBr46] and I. J. Good [Go46] in 1946, and have played important roles in both pure mathematics (for example, enumeration combinatorics) and applied mathematics (for example, genome assembly). The appearance of de Bruijn–Good directed graphs in this paper is natural, as walks on these graphs correspond to the orbits under the shift operator $\sigma$ (see [BZ16]).

The investigations on the directed cycle with the maximum mean-weight in a directed graph is sometimes known as the maximum mean cycle problem. This problem is one of the most fundamental problems in combinatorial optimization and has a long history of algorithmic studies, for examples [DIG99, Ka78].

It is also worth to point out that our methods should be extendable to shifts on an arbitrary finite number of symbols or more generally to subshifts of finite type by investigating natural generalizations of de Bruijn–Good directed graphs. Favoring clarity at the expense of generality, we opted to work with the full shift on $2$ symbols. In this context, we are able to express with relative ease the random perturbations key to the proof of Theorem 1.1.

The paper is organized as follows. Section 2 is devoted to preliminaries, where we recall some basic facts about full shifts, Haar series, and the compactness of Hilbert bricks (see Lemma 2.1). Section 3 provides a more concrete formulation (see Theorem 3.1) of our main result Theorem 1.1. In Section 4 we recall the de Bruijn–Good directed graphs and the gap criterion for the locking property that was developed in [BZ16], see Lemma 4.1. In Section 5 we adapt a strategy from [CCK15, Section 5] to make refined estimates for the (conditional) gap between the mean-weights of the two heaviest directed cycles, see Proposition 5.1 and Proposition 5.2. Using these estimates, we show in Section 6 that hypothesis (1.3) guarantees that with probability $1$, the gap criterion is satisfied at some finite step in the finite-dimensional approximations. Thus we complete the proof of Theorem 1.1. Finally, Section  7 is devoted to the proofs of Theorems 1.2 and 1.3.

Acknowledgments

Y. Zhang is grateful to Peking University for the hospitality, during his stay, when part of this work was done. J. Ding was partially supported by NSFC Key Program Project No. 12231002. Z. Li was partially supported by NSFC Nos. 12101017, 12090015, and BJNSF No. 1214021. Y. Zhang was partially supported by NSFC Nos. 12161141002, 11871262, and 12271432.

2. Preliminaries

In this section, we will introduce the background and necessary concepts.

2.1. Full shift

We follow the convention that $\mathbb{N}\coloneqq\{1,\,2,\,3,\,\dots\}$ and $\mathbb{N}_{0}\coloneqq\{0\}\cup\mathbb{N}$ in this paper.

For each $n\in\mathbb{N}$, the set $\Sigma_{n}\coloneqq\{0,\,1\}^{n}$ of words of length $n$ (or $n$-words) in the alphabet $\{0,\,1\}$ is the set of binary sequences of length $n$. Write $\Sigma_{0}\coloneqq\{\emptyset\}$ and call $\emptyset$ the word of length [math]. Define $\lvert\bm{w}\rvert\coloneqq n$ for each $n$-word, $n\in\mathbb{N}_{0}$. The set $\Sigma^{+}\coloneqq\{0,\,1\}^{\mathbb{N}}$ of infinite words in the alphabet $\{0,\,1\}$ is the set of one-sided infinite binary sequences. Denote the set of finite words by $\Sigma_{*}\coloneqq\bigcup_{n\in\mathbb{N}_{0}}\Sigma_{n}$ and the set of words of length at most $k$, for $k\in\mathbb{N}_{0}$, by $\Sigma_{\leq k}\coloneqq\bigcup_{n=0}^{k}\Sigma_{n}$. Note that $\Sigma_{*}$ includes $\emptyset$, i.e., the word of length [math]. We often write $\{w_{n}\}_{n\in\mathbb{N}}=w_{1}w_{2}\dots w_{n}\dots$ for an infinite word $\bm{w}=\{w_{n}\}_{n\in\mathbb{N}}\in\Sigma^{+}$, and similarly, write $\{z_{n}\}_{n=1}^{m}=z_{1}z_{2}\dots z_{m}$ for an $m$-word $\bm{z}=\{z_{n}\}_{n=1}^{m}\in\Sigma_{m}$, $m\in\mathbb{N}$.

The left-shift operator $\sigma\colon\Sigma^{+}\rightarrow\Sigma^{+}$ is given by

[TABLE]

The pair $(\Sigma^{+},\sigma)$ is called the full shift on $\Sigma^{+}=\{0,\,1\}^{\mathbb{N}}$.

For each pair of finite words $\bm{w},\,\bm{z}\in\Sigma_{*}$, define $\bm{w}\bm{z}$ to be their concatenation if $\bm{w}\neq\emptyset$ and set $\bm{w}\bm{z}\coloneqq\bm{z}$ otherwise, and denote by $[\bm{w}]\coloneqq\{\bm{w}\bm{y}:\bm{y}\in\Sigma^{+}\}$ the cylinder set beginning at $\bm{w}$.

An infinite word $\bm{w}\in\Sigma^{+}$ is a periodic point of $(\Sigma^{+},\sigma)$ if $\sigma^{n}(\bm{w})=\bm{w}$ for some $n\in\mathbb{N}$, and the smallest such positive integer $n$ is called the period of $\bm{w}$. Let $P_{\sigma}$ denote the set of all periodic points of $(\Sigma^{+},\sigma)$.

Let $\bm{x}=\{x_{i}\}_{i\in\mathbb{N}_{0}}$ and $\bm{y}=\{y_{i}\}_{i\in\mathbb{N}_{0}}$ be two elements of $\Sigma^{+}$. Denote by $\bm{x}\mathbin{\dagger}\bm{y}$ the position of first disagreement between the sequences $\bm{x}$ and $\bm{y}$, that is, the least $m\in\mathbb{N}$ such that $x_{m}\neq y_{m}$, with the convention $\bm{x}\mathbin{\dagger}\bm{x}=+\infty$. The following properties hold:

[TABLE]

For each $n\in\mathbb{N}$, we define the $n$-th variation of a function $f\colon\Sigma^{+}\rightarrow\mathbb{R}$ as:

[TABLE]

2.2. Haar series

As mentioned in [BZ16, Section 1.3], for each $\bm{w}\in\Sigma_{*}$, we define the Haar function:

[TABLE]

These functions are straightforward adaptations of the classical Haar functions on $[0,1]$ (see for example, [Pi09, Section 6.3]). It has an advantage to work on the Cantor set $\Sigma^{+}$ over working on the interval $[0,1]$: the Haar functions defined in (2.2) are continuous. We follow the choice of the normalization in [BZ16], which makes subsequent formulas simpler.

If a function $f\colon\Sigma^{+}\rightarrow\mathbb{R}$ satisfies $\mbox{var}_{n}(f)=0$ for some $n\in\mathbb{N}_{0}$, i.e., it is constant in each cylinder of level $n$, then it is called a step function of level $n$. Such functions form a vector space $\mathcal{S}_{n}$ of dimension $2^{n}$. For each $n\in\mathbb{N}$, the set $\{1\}\cup\{h_{\bm{w}}:\lvert\bm{w}\rvert<n\}$ forms a basis of the vector space $\mathcal{S}_{n}$.

Let $\beta$ be the unbiased Bernoulli measure on $\Sigma^{+}$, i.e., the probability measure that assigns equal weights to all cylinders of the same level. Let $L^{2}(\beta)$ denote the Hilbert space of functions that are square-integrable with respect to the measure $\beta$. Then the set $\{1\}\cup\{h_{\bm{w}}:\bm{w}\in\Sigma_{*}\}$ is an orthogonal basis of $L^{2}(\beta)$. Thus every $f\in L^{2}(\beta)$ can be represented by a Haar series:

[TABLE]

where the Haar coefficients are defined as:

[TABLE]

for each $\bm{w}\in\Sigma_{n}$. It follows from [BZ16, Equations (2.2), (2.3), and (2.4)] that the $n$-th approximation $A_{n}f$ of $f\in L^{2}(\beta)$ has the following equivalent characterizations:

• •

the projection of $f$ on the subspace $\mathcal{S}_{n}$ along the orthogonal complement of $\mathcal{S}_{n}$;

• •

the sum of the truncated Haar series:

[TABLE]

• •

the function obtained by averaging $f$ on cylinders of level $n$:

[TABLE]

Thus we have

[TABLE]

which, under the additional assumption that $f$ is continuous, converges to [math] as $n$ tends to $+\infty$. Here we use the notation in (2.1). Therefore every continuous function $f$ can be written as a uniformly convergent series:

[TABLE]

which, with some abuse of language, we also call a Haar series.

2.3. Gauges

Let $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ be a strictly decreasing sequence of positive numbers converging to [math].

Consider a function $f\in C(\Sigma^{+})$. It follows from (2.1) and (2.3) that for each $n\in\mathbb{N}$ and each $\bm{w}\in\Sigma_{n}$, we have

[TABLE]

Similarly, if $f$ is Lipschitz with respect to $d_{\bm{a}}$, then

[TABLE]

A gauge is a family $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ of positive numbers indexed by finite words. We say that a gauge $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ is weakly $\bm{a}$-admissible if

[TABLE]

where

[TABLE]

The proof of [BZ16, Lemma 1.1] in [BZ16, Section 2] establishes the following lemma, which is stronger than the statement of [BZ16, Lemma 1.1].

Lemma 2.1**.**

Let $(\Sigma^{+},\sigma)$ be the full shift on binary sequences. Suppose that $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ is a strictly decreasing sequence of positive numbers converging to zero satisfying $\sum_{k\geq n}a_{k}=O(a_{n})$ as $n\to+\infty$. Let $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ be a weakly $\bm{a}$-admissible gauge. For a family $\bm{e}=\{e_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ of real numbers satisfying $\lvert e_{\bm{w}}\rvert\leq b_{\bm{w}}$ for each $\bm{w}\in\Sigma_{*}$, the Haar series $\sum_{\bm{w}\in\Sigma_{*}}e_{\bm{w}}h_{\bm{w}}$ converges in the uniform norm to a unique function $f_{\bm{e}}$ in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. Moreover, the set

[TABLE]

is a compact subset of $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$.

The set $\mathcal{H}_{\bm{b}}$ as in the lemma above is called the Hilbert brick with gauge $\bm{b}$. By means of Haar coefficients, it can be identified with the product space $\prod_{\bm{w}\in\Sigma_{*}}[-b_{\bm{w}},b_{\bm{w}}]$. In particular, we can endow each interval $[-b_{\bm{w}},b_{\bm{w}}]$, $\bm{w}\in\Sigma_{*}$, with a probability measure $\mu_{\bm{w}}$. Then by using the identification from Lemma 2.1, we obtain a probability measure $\mathbb{P}_{\bm{b}}\coloneqq\prod_{\bm{w}\in\Sigma_{*}}\mu_{\bm{w}}$ on the Banach space $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ supported on the compact subset $\mathcal{H}_{\bm{b}}$.

In this paper, we are most interested in the special case when all $\mu_{\bm{w}}$ are uniform measures on $[-b_{\bm{w}},b_{\bm{w}}]$.

3. A concrete formulation of Theorem 1.1

Theorem 1.1 follows from an explicit construction of a compactly supported probability measure that is transverse to the compliment of $\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$. Therefore, we will formulate Theorem 1.1 in terms of Theorem 3.1, which has concrete information on such a measure.

Theorem 3.1**.**

Let $(\Sigma^{+},\sigma)$ be the full shift on binary sequences. Let $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ be a strictly decreasing sequence of positive numbers satisfying $a_{n+1}/a_{n}=O(\theta^{n})$ as $n\to+\infty$ for some $\theta\in(0,1/4)$. Set $b_{n}\coloneqq a_{n}/n$ for each $n\in\mathbb{N}_{0}$. Consider the weakly $\bm{a}$-admissible gauge $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ given by $b_{\bm{w}}\coloneqq b_{\lvert\bm{w}\rvert}$ for each $\bm{w}\in\Sigma_{*}$. Then there exists a probability measure $\mathbb{P}_{\bm{b}}$ supported on the compact set $\mathcal{H}_{\bm{b}}$ such that for each $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$,

[TABLE]

and consequently, $\mathbb{P}_{\bm{b}}(\{g\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}}):f_{0}+g\notin\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})\})=0$.

A proof of Theorem 3.1 will be presented in Section 6.

Theorem 3.1 concludes that the measure $\mathbb{P}_{\bm{b}}$ is transverse to the complement of the set $\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ and is supported on the compact set $\mathcal{H}_{\bm{b}}$. Thus $\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in $\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. Hence Theorem 3.1 implies Theorem 1.1.

4. A gap criterion for the locking property

From now on, our aim is to establish Theorem 3.1. To this end, we need to utilize a criterion for the locking property in terms of some probabilistic objects for the maximum mean cycle problem on de Bruijn–Good digraphs.

Let us make some remarks to explain the relations between (infinite dimensional) ergodic optimization, finite dimensional ergodic optimization, and the maximum mean cycle problem on digraphs. The facts stated below until the start of Subsection 4.1 serve merely to assist the reader in comprehending our approach and will not be used in the proofs.

Recall that $P_{\sigma}$ denotes the set of all periodic points of $(\Sigma^{+},\sigma)$. Each periodic orbit in $P_{\sigma}$ is associated to a unique invariant probability measure supported on it. It is well-known that the set of these measures is a dense subset in $\mathcal{M}(\Sigma^{+},\sigma)$. In fact, $\mathcal{M}(\Sigma^{+},\sigma)$ is a Poulsen simplex introduced by E. T. Poulsen [Po61], which has a dense set of extreme points. Consider a sequence of maps $\{\pi_{n}\}_{n\in\mathbb{N}_{0}}$ where $\pi_{n}\colon\mathcal{M}(\Sigma^{+},\sigma)\rightarrow\mathbb{R}^{2^{n}}$ is given by $\pi_{n}(\mu)\coloneqq\{\mu([\bm{w}])\}_{\bm{w}\in\Sigma_{n}}$ for $\mu\in\mathcal{M}(\Sigma^{+},\sigma)$. Denote $R_{n}\coloneqq\pi_{n}(\mathcal{M}(\Sigma^{+},\sigma))$. It then follows from Sections 3 and 4 in [Zi95] (see also Subsection 3.5 in [BZ16]) that $R_{0}\subset R_{1}\subset\cdots$ form a nested sequence of finite dimensional polyhedra, whose vertices are measures supported on some periodic orbits in $P_{\sigma}$. Moreover, each polyhedron $R_{n}$ is a projection of the next one $R_{n+1}$, and $\mathcal{M}(\Sigma^{+},\sigma)$ can be recovered as the inverse limit of the sequence $\{R_{n}\}_{n\in\mathbb{N}_{0}}$.

Since $\mathcal{M}(\Sigma^{+},\sigma)$ is a Poulsen simplex, the polyhedra in $\{R_{n}\}_{n\in\mathbb{N}_{0}}$ are not regular simplices. On the contrary, they have huge numbers of vertices, and their faces are small. By Proposition 3.5 in [BZ16], $R_{n}$ is isomorphic to the so-called “circulation polytope” of the de Bruijn–Good digraph $\operatorname{\mathcal{BG}}_{n}$. With the aid of this isomorphism, the finite dimensional ergodic optimization problem over $R_{n}$ can be conveniently restated as the maximum mean cycle problem on the de Bruijn–Good digraph $\operatorname{\mathcal{BG}}_{n}$, and a criterion for the locking property is expressed in terms of the gap between the mean-weights of two heaviest directed cycles in the de Bruijn–Good digraph accordingly, see the gap criterion in Subsection 4.2. This step requires some notions from graph theory.

4.1. Graph theory

In this subsection, we go over some key concepts and notations in graph theory. For a more detailed introduction to graph theory, we refer to [BM08].

A directed graph (or digraph) $\mathcal{D}$ is an ordered pair $(V(\mathcal{D}),A(\mathcal{D}))$ of sets, where $V(\mathcal{D})$ is the set of vertices and $A(\mathcal{D})$ is the set of arcs, together with an incidence function $\psi_{\mathcal{D}}\colon A(\mathcal{D})\rightarrow V(\mathcal{D})\times V(\mathcal{D})$ that sends an arc $a$ to an ordered pair $(u,v)$ of vertices. For an arc $a\in A(\mathcal{D})$ with $\psi(a)=(u,v)$, we call the vertex $u$ the tail of $a$, denoted by $\operatorname{tail}(a)$, and call $v$ the head of $a$, denoted by $\operatorname{head}(a)$. With abuse of language, we use “arc” and “edge” interchangeably in this paper.

A digraph $\mathcal{D}^{\prime}$ is a subdigraph of a digraph $\mathcal{D}$ if $V(\mathcal{D}^{\prime})\subseteq V(\mathcal{D})$, $A(\mathcal{D}^{\prime})\subseteq A(\mathcal{D})$, and $\psi_{\mathcal{D}^{\prime}}$ is the restriction of $\psi_{\mathcal{D}}$ to $A(\mathcal{D}^{\prime})$. We then say that $\mathcal{D}$ contains $\mathcal{D}^{\prime}$ or that $\mathcal{D}^{\prime}$ is contained in $\mathcal{D}$, and write $\mathcal{D}\supseteq\mathcal{D}^{\prime}$ or $\mathcal{D}^{\prime}\subseteq\mathcal{D}$, respectively.

A directed cycle $\mathcal{C}$ is a digraph satisfying that $\operatorname{card}{A(\mathcal{C})}=\operatorname{card}{V(\mathcal{C})}\eqqcolon n\in\mathbb{N}$, that $V(\mathcal{C})=\bigcup\{\{\operatorname{tail}(a),\,\operatorname{head}(a)\}:a\in A(\mathcal{C})\}$, and that there is an enumeration $a_{0},\,a_{1},\,\dots,\,a_{n-1}$ of $A(\mathcal{C})$ such that $\operatorname{head}(a_{i})=\operatorname{tail}(a_{i+1})$ for each $i\in\{0,\,1,\,\dots,\,n-1\}$, where $a_{n}\coloneqq a_{0}$. For a directed cycle $\mathcal{C}$ we write $\operatorname{len}(\mathcal{C})$ for the length (i.e., the number of arcs) of $\mathcal{C}$. For a digraph $D$, we denote by $\mathscr{C}(D)$ the collection of all directed cycles contained in $D$.

The de Bruijn–Good digraph $\operatorname{\mathcal{BG}}_{n}$, $n\in\mathbb{N}$, is a digraph whose set of vertices $V(\operatorname{\mathcal{BG}}_{n})\coloneqq\Sigma_{n-1}$ and set of arcs $A(\operatorname{\mathcal{BG}}_{n})\coloneqq\Sigma_{n}$ consists of words of length $n-1$ and words of length $n$, respectively, which satisfies that $\operatorname{tail}(w_{1}\dots w_{n})=w_{1}\dots w_{n-1}$ and $\operatorname{head}(w_{1}\dots w_{n})=w_{2}\dots w_{n}$ for each $n$-word $w_{1}\dots w_{n}\in A(\operatorname{\mathcal{BG}}_{n})$. The first five de Bruijn–Good digraphs are shown in Figure 4.1.

4.2. Gap criterion

Fix a real-valued continuous function $f\in C(\Sigma^{+})$ and an integer $n\in\mathbb{N}$. We define inductively weights $W_{n}^{f}(\bm{w})$ associated to $f$ on the edges $\bm{w}$ of the de Bruijn–Good digraph $\operatorname{\mathcal{BG}}_{n}$ as follows:

• •

If $n=1$, then $W_{1}^{f}(\alpha)\coloneqq\frac{(-1)^{\alpha}}{2}c_{\emptyset}(f)$ for each $\alpha\in\{0,1\}$;

• •

If $n\geq 2$, then $W_{n}^{f}(\bm{w}\alpha)\coloneqq W_{n-1}^{f}(\bm{w})+\frac{(-1)^{\alpha}}{2}(c_{\bm{w}}(f))$ for all $\bm{w}\in\Sigma_{n-1}$ and $\alpha\in\{0,1\}$.

Thus

[TABLE]

The pair of \bigl{(}\operatorname{\mathcal{BG}}_{n},W_{n}^{f}\bigr{)} induces the mean-weight (induced by $f$)

[TABLE]

on each directed cycle $\mathcal{C}$ contained in $\operatorname{\mathcal{BG}}_{n}$.

Denote by $\operatorname{Gap}_{n}(f)\coloneqq M_{n}^{f}(\mathcal{C}_{1})-M_{n}^{f}(\mathcal{C}_{2})\geq 0$ the gap in the mean-weights of the two heaviest (in terms of mean-weight) directed cycles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ contained in $\operatorname{\mathcal{BG}}_{n}$.

It is easy to check that

[TABLE]

for $\bm{w}\in\Sigma_{n}$ and $\mathcal{C}\in A(\operatorname{\mathcal{BG}}_{n})$.

In the notations above, we have the following gap criterion from [BZ16, Lemma 4.2].

Lemma 4.1** (Gap criterion for the locking property [BZ16]).**

Let $(\Sigma^{+},\sigma)$ be the full shift on binary sequences. Consider a strictly decreasing sequence $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ of positive numbers satisfying

[TABLE]

If a function $f\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ satisfies the the following inequality:

[TABLE]

for some $n\in\mathbb{N}$, then $f\in\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$.

In informal terms, if the tail of the Haar series is small compared to the gap of its initial part (upto the level $n-1$ (see (2.4), (4.1), and (4.5))), then it does not influence the maximizing measure. Note that $f\in\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ then in particular $f\in\mathscr{P}\mathscr{U}(\sigma)$.

5. Random perturbations on the edge-weight

In this section, we will describe the random weights on edges of the de Bruijn–Good digraphs $\operatorname{\mathcal{BG}}_{n}$, $n\in\mathbb{N}$. Namely, we deal with the Haar expansion of $f_{0}+g$, with $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ (the deterministic part) and $g\in\mathcal{H}_{\bm{b}}$ (the random part).

Let $\{Y_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ be a collection of i.i.d. random variables, having the uniform distribution on $[-1,1]$, indexed by the set $\Sigma_{*}$ of finite words.

Suppose that $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ is a strictly decreasing sequence of positive numbers converging to zero satisfying $\sum_{k\geq n}a_{k}=O(a_{n})$ as $n\to+\infty$. Let $b_{n}$, $n\in\mathbb{N}_{0}$, be a sequence of positive numbers satisfying $b_{n}=o(a_{n})$ as $n$ tends to $+\infty$.

We formally define a random function

[TABLE]

which has random coefficients on its Haar series. By Lemma 2.1, the series above always converge uniformly, and the limit $g$ is Lipschitz with respect to $d_{\bm{a}}$. For each $n\in\mathbb{N}$, the $n$-th approximation $A_{n}g$ of $g$ is given by

[TABLE]

Fix an arbitrary function $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. With abuse of notation, we use similar symbols for the random edge-weight, mean-weight, and gap induced by random functions that are continuous. In particular, the (random) edge-weights $W_{n}^{f_{0}+g}(\bm{w})$, $n\in\mathbb{N}$, on $\operatorname{\mathcal{BG}}_{n}$ induced by $f_{0}+g$ are defined by the following inductive process:

• •

If $n=1$, then $W_{1}^{f_{0}+g}(\alpha)=\frac{(-1)^{\alpha}}{2}(c_{\emptyset}(f_{0})+b_{0}\cdot Y_{\emptyset})$ for each $\alpha\in\{0,1\}$;

• •

If $n\geq 2$, then $W_{n}^{f_{0}+g}(\bm{w}\alpha)=W_{n-1}^{f_{0}+g}(\bm{w})+\frac{(-1)^{\alpha}}{2}(c_{w_{1}w_{2}\cdots w_{n-1}}(f_{0})+b_{n-1}\cdot Y_{\bm{w}})$ for all $\bm{w}\in\Sigma_{n-1}$ and $\alpha\in\{0,1\}$.

Fix an integer $n\in\mathbb{N}$. We then have

[TABLE]

for $\bm{w}\in\Sigma_{n}$. Moreover, the (random) mean-weight of a directed cycle $\mathcal{C}$ in $\operatorname{\mathcal{BG}}_{n}$ induced by $f_{0}+g$ is then given by

[TABLE]

Finally, the (random) gap, induced by $f_{0}+g$, between the mean-weights of the two heaviest (in terms of mean-weight) directed cycles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ in $\operatorname{\mathcal{BG}}_{n}$ is

[TABLE]

Since uniform measures are continuous, almost surely there is a unique directed cycle with the maximal mean-weight, i.e., $\operatorname{Gap}_{n}(f_{0}+g)>0$.

The main technical result is formulated below.

Proposition 5.1**.**

Let $(\Sigma^{+},\sigma)$ be the full shift on binary sequences. Let $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ be a strictly decreasing sequence of positive numbers satisfying $a_{n+1}/a_{n}=O(\theta^{n})$ as $n\to+\infty$ for some $\theta\in(0,1/2)$. Let $b_{n}$, $n\in\mathbb{N}_{0}$, be a sequence of positive numbers satisfying $b_{n}=o(a_{n})$ as $n\to+\infty$. Consider a weakly $\bm{a}$-admissible gauge $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ satisfying $b_{\bm{w}}=b_{\lvert\bm{w}\rvert}$ for all $\bm{w}\in\Sigma_{*}$. Fix an arbitrary function $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. Consider the random function $g$ defined in (5.1).

Consider an integer $n\geq 2$, a real number $\epsilon\in(0,1/2)$, and a collection $\bm{t}=\{t_{\bm{w}}\}_{{\bm{w}}\in\Sigma_{\leq n-2}}$ of real numbers $t_{\bm{w}}\in\mathbb{R}$. Let $\{X_{\bm{t}}(\mathcal{C}):\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})\}$ be a collection of random variables with joint distribution given by the conditional distribution of \bigl{\{}M_{n}^{f_{0}+g}(\mathcal{C}):\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})\bigr{\}} given $\{Y_{\bm{w}}=t_{\bm{w}}:\bm{w}\in\Sigma_{\leq n-2}\}$. Let $\operatorname{Gap}_{n,\bm{t}}$ be the absolute value of the difference between the largest two values in $\{X_{\bm{t}}(\mathcal{C}):\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})\}$. Then

[TABLE]

Note that $A(\mathcal{C}_{1})\nsubseteq A(\mathcal{C}_{2})$ for each pair of distinct directed cycles contained in a directed graph $\mathcal{D}$. Then by (5.3), (5.4), and the definition of $Y_{\bm{w}}$ for $\bm{w}\in\Sigma_{*}$, we get that the random functions $X_{\bm{t}}(\mathcal{C})$ and $X_{\bm{t}}(\mathcal{C}^{\prime})$ are distinct linear combinations of random variables $Y_{\bm{w}}$, $\bm{w}\in\Sigma_{\leq n-1}$, (with scalar multiples of Haar functions as coefficients) for distinct directed cycles $\mathcal{C}$ and $\mathcal{C}^{\prime}$ contained in $\operatorname{\mathcal{BG}}_{n}$. Moreover, $\operatorname{Gap}_{n,\bm{t}}$ in the proposition is well-defined almost surely.

Proof.

Consider $n$, $\epsilon$, and $\bm{t}$ as given in the hypothesis.

Let $\mathcal{E}$ be the event that $\lvert Y_{\bm{w}}\rvert\leq 1-2\epsilon$ for all $\bm{w}\in\Sigma_{n-1}$.

We will investigate, for each $\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$, the standard deviation $\sigma_{\mathcal{C}}\coloneqq\sigma(X_{\bm{t}}(\mathcal{C}))$, the mean $\mu_{\mathcal{C}}\coloneqq\mathbb{E}[X_{\bm{t}}(\mathcal{C})]$, the probability density function $\phi_{\mathcal{C}}$ for $X_{\bm{t}}(\mathcal{C})$, and the probability density function $\widetilde{\phi}_{\mathcal{C}}$ for $X_{\bm{t}}(\mathcal{C})$ restricted to $\mathcal{E}$. Then it follows from the definition of $X_{\bm{t}}$, the fact that $\mathbb{E}[Y_{\bm{w}}]=0$ for $\bm{w}\in\Sigma_{*}$, (5.3), and (5.4) that

[TABLE]

for each $\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$. Here and henceforth by $z_{\mathcal{C}}(\bm{w})$ we denote, for each $\bm{w}\in V(\mathcal{C})$, the unique number in $\{0,\,1\}$ such that $\bm{w}z_{\mathcal{C}}(\bm{w})\in A(\mathcal{C})$. Note that

[TABLE]

Define, for each $\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$ and each pair of numbers $x,\,x^{\prime}\in\mathbb{R}$, the event

[TABLE]

and the conditional probabilities

[TABLE]

We first verify the following two claims.

Claim 1. $\phi_{\mathcal{C}}(x+\Delta)\geq\widetilde{\phi}_{\mathcal{C}}(x)$ for all $\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$, $\Delta\in[0,b_{n-1}\epsilon]$, and $x\in\mathbb{R}$ satisfying $\lvert x-\mu_{\mathcal{C}}\rvert\leq 1-b_{n-1}\epsilon$.

To establish Claim 1, we fix $\mathcal{C}$, $\Delta$, and $x$ as in the claim. We consider a collection $\{y_{\bm{w}}\}_{\bm{w}\in V(\mathcal{C})}$ of real numbers satisfying both

[TABLE]

and $\lvert y_{\bm{w}}\rvert\leq 1-2\epsilon$ for each $\bm{w}\in V(\mathcal{C})$. We set

[TABLE]

for each $\bm{w}\in V(\mathcal{C})$. Then

[TABLE]

for each $\bm{w}\in V(\mathcal{C})$, and

[TABLE]

Claim 1 then follows from (5.7) and the translation invariance of uniformly distributed random variables.

By a similar argument as in the proof of Claim 1, we get the following claim.

Claim 2. Q_{\mathcal{C}}\bigl{(}x+\Delta,\,x+\bigl{(}1-2^{-n+1}\bigr{)}\Delta\bigr{)}\geq\widetilde{Q}_{\mathcal{C}}(x) for all $\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$, $\Delta\in[0,b_{n-1}\epsilon]$, and $x\in\mathbb{R}$ satisfying $\lvert x-\mu_{\mathcal{C}}\rvert\leq 1-b_{n-1}\epsilon$.

To establish Claim 2, we fix $\mathcal{C}$, $\Delta$, and $x$ as in the claim. We consider a collection $\{y_{\bm{w}}\}_{\bm{w}\in\Sigma_{n-1}}$ of real numbers satisfying both the equality in (5.9) and $\lvert y_{\bm{w}}\rvert\leq 1-2\epsilon$ for each $\bm{w}\in\Sigma_{n-1}$. We set

[TABLE]

for each $\bm{w}\in V(\mathcal{C})$, and set $y^{\prime}_{\bm{u}}\coloneqq y_{\bm{u}}$ for each $\bm{u}\in\Sigma_{n-1}\setminus V(\mathcal{C})$. Then $\lvert y^{\prime}_{\bm{w}}\rvert\leq 1$ for all $\bm{w}\in\Sigma_{n-1}$ as

[TABLE]

for each $\bm{w}\in V(\mathcal{C})$. On the other hand, consider an arbitrary directed cycle $\mathcal{C}^{\prime}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})\setminus\{\mathcal{C}\}$. Since $\mathcal{C}^{\prime}$ and $\mathcal{C}$ are distinct directed cycles, it is clear that $A(\mathcal{C}^{\prime})\setminus A(\mathcal{C})\neq\emptyset$. Consequently,

[TABLE]

Claim 2 then follows from (5.7) and the translation invariance of uniformly distributed random variables.

Finally, we apply the two claims to finish the proof of the proposition.

Provided $x^{\prime}<x$, we define the events $\mathcal{Q}(\mathcal{C})\coloneqq\mathcal{R}_{\bm{t}}(\mathcal{C},x^{\prime})\cap\{X_{\bm{t}}(\mathcal{C})=x\}$, $\mathcal{C}\in\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$. We observe that $\mathcal{Q}(\mathcal{C})$ and $\mathcal{Q}(\mathcal{C}^{\prime})$ are disjoint for distinct directed cycles $\mathcal{C}$ and $\mathcal{C}^{\prime}$ in $\mathscr{C}(\operatorname{\mathcal{BG}}_{n})$.

Define $\iota\coloneqq b_{n-1}\epsilon$. Then by the two claims, the observation above, (5.7), (5.8), and the definitions of $Q_{\mathcal{C}}$, $\widetilde{Q}_{\mathcal{C}}$, $\mathcal{R}_{\bm{t}}$, and $\mathcal{E}$, we get

[TABLE]

We remark that in the last inequality, we used the fact that $\widetilde{\phi}_{\mathcal{C}}$ is zero outside of $[\mu_{\mathcal{C}}-2^{-1}b_{n-1},\mu_{\mathcal{C}}+2^{-1}b_{n-1}]$ by (5.8).

As a straightforward consequence of the uniform distribution and the hypothesis that $\epsilon\in(0,1/2)$, we have

[TABLE]

where the inequality follows from the well-known inequality $1-m\alpha\leq(1-\alpha)^{m}$ for $m\in\mathbb{N}$ and $\alpha\in(0,1)$ from the convexity of the logarithm. The proposition is therefore established. ∎

The following proposition brings us a step closer to the gap criterion in Lemma 4.1.

Proposition 5.2**.**

Let $(\Sigma^{+},\sigma)$ be the full shift on binary sequences. Let $\bm{a}=\{a_{n}\}_{n\in\mathbb{N}_{0}}$ be a strictly decreasing sequence of positive numbers satisfying $a_{n+1}/a_{n}=O(\theta^{n})$ as $n\to+\infty$ for some $\theta\in(0,1/2)$. Let $b_{n}$, $n\in\mathbb{N}_{0}$, be a sequence of positive numbers satisfying $b_{n}=o(a_{n})$ as $n\to+\infty$. Consider a weakly $\bm{a}$-admissible gauge $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ satisfying $b_{\bm{w}}=b_{\lvert\bm{w}\rvert}$ for all $\bm{w}\in\Sigma_{*}$.

Then there exists a measure $\mathbb{P}_{\bm{b}}$ supported on $\mathcal{H}_{\bm{b}}$ with the property that for every $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$, we have

[TABLE]

Proof.

Fix an arbitrary integer $n\geq 2$.

It follows from Proposition 5.1 that

[TABLE]

for all $\epsilon\in(0,1/2)$.

We denote by $\phi_{n-2}$ the joint probability density function for all the random variables in $\{Y_{\bm{w}}:\bm{w}\in\Sigma_{\leq n-2}\}$. Then for each $\epsilon\in(0,1/2)$,

[TABLE]

Taking $\epsilon\coloneqq 2^{-n}n^{-3}$, then , we have

[TABLE]

By the Borel–Cantelli lemma, therefore, the proposition follows. ∎

6. Proof of Theorem 3.1

We are now ready to establish the more concrete formulation Theorem 3.1 of our main theorem (Theorem 1.1).

Proof of Theorem 3.1.

Fix an arbitrary function $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. Define

[TABLE]

Next, for each $g\in\Lambda_{f_{0}}$, by the definition of $\Lambda_{f_{0}}$, for all sufficiently large $n\in\mathbb{N}$,

[TABLE]

On the other hand, since by the hypothesis on $\bm{a}$ and $\bm{b}$,

[TABLE]

and since $\theta\in(0,1/4)$, for all sufficiently large integer $n$, one has

[TABLE]

and

[TABLE]

Combining (6.1), (6.3), $b_{n}=a_{n}/n$, (6.4), (2.6), (2.7), and the triangle inequality, we deduce that for each $N\in\mathbb{N}$, there exists an integer $n\geq N$ such that

[TABLE]

Note that by (2.6) and (2.7), $a_{k}\operatorname{LIP}_{d_{\bm{a}}}(f_{0})\geq\max_{\bm{w}\in\Sigma_{k}}\{\lvert c_{\bm{w}}(f_{0})\rvert\}$. On the other hand, since $g\in\Lambda_{f_{0}}$, by the definition of $\Lambda_{f_{0}}$ and Lemma 2.1, we have $b_{k}\geq\max_{\bm{w}\in\Sigma_{k}}\{\lvert c_{\bm{w}}(g)\rvert\}$. Hence

[TABLE]

This means that $f_{0}+g$ satisfies the gap criterion (4.7) at level $n$. Together with (6.2) and Lemma 4.1 (on $f_{0}+g$), it then follows that $f_{0}+g\in\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$.

Hence for each $f_{0}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ we have

[TABLE]

Then it follows from Proposition 5.2,

[TABLE]

The proof of Theorem 3.1 is therefore complete. ∎

7. Proofs of Theorem 1.2 and Theorem 1.3

Recall the definition of little Lipschitz spaces reviewed in Section 1.

Proof of Theorem 1.2.

It suffices to show that $\operatorname{lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$.

Set $b_{n}\coloneqq a_{n}/n$ for each $n\in\mathbb{N}_{0}$. Consider the weakly $\bm{a}$-admissible gauge $\bm{b}=\{b_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ given by $b_{\bm{w}}\coloneqq b_{\lvert\bm{w}\rvert}$ for each $\bm{w}\in\Sigma_{*}$.

We first show that the Hilbert brick $\mathcal{H}_{\bm{b}}$ defined in (2.9) in Lemma 2.1 is a subset of $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$.

Indeed, for each $\bm{e}=\{e_{\bm{w}}\}_{\bm{w}\in\Sigma_{*}}$ with $\lvert e_{\bm{w}}\rvert\leq b_{\bm{w}}$ for each $\bm{w}\in\Sigma_{*}$, by Lemma 2.1 the uniform limit $f_{\bm{e}}\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ of the Haar series $\sum_{\bm{w}\in\Sigma_{*}}e_{\bm{w}}h_{\bm{w}}$ satisfies that for each $n\in\mathbb{N}$ sufficiently large and each pair of $\bm{x},\,\bm{y}\in\Sigma^{+}$ with $\bm{x}\mathbin{\dagger}\bm{y}=n$, by (2.2), the hypothesis on $\bm{a}$, and our choice of $b_{n}$, we have

[TABLE]

Thus by the definitions of the metric $d_{\bm{a}}$ and of the little Lipschitz space, we have $f_{\bm{e}}\in\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$. It follows from (2.9) that $\mathcal{H}_{\bm{b}}\subseteq\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$.

Note that $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})\subseteq\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$ and that

[TABLE]

Hence by Theorem 3.1, the probability measure $\mathbb{P}_{\bm{b}}$ from Theorem 3.1 supported on the compact set $\mathcal{H}_{\bm{b}}$ satisfies that for each $f_{0}\in\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$,

[TABLE]

Therefore, $\operatorname{lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ is a prevalent set in $\operatorname{lip}(\Sigma^{+},d_{\bm{a}})$, and the theorem is established. ∎

The deduction of Theorem 1.3 from Theorems 1.1 and 1.2 is not difficult. We include it for the convenience of the reader.

Proof of Theorem 1.3.

We refer the reader to [PU10] for relevant notions from ergodic theory used in this proof. Observe that $\sigma$ is open, uniformly distance expanding, and topologically exact on $(\Sigma^{+},d_{\bm{a}})$.

Consider a potential $f\in\operatorname{Lip}(\Sigma^{+},d_{\bm{a}})$. By [PU10, Theorem 5.6.2], there exists a unique equilibrium state $\mu_{tf}$ for each $t\in\mathbb{R}$. We get from the definition of equilibrium states (in Section 1) and (1.1) that

[TABLE]

for all $\nu\in\mathcal{M}(\Sigma^{+},\sigma)$. Note that the topological entropy $h_{\operatorname{top}}(\sigma)=\log 2$ provides an upper bound for the measure-theoretic entropy $h_{\mu}(\sigma)$. If $\mu$ is a ground state, i.e., $\mu=\lim\limits_{i\to+\infty}\mu_{t_{i}f}$ in the weak∗ topology for some sequence $t_{i}$, $i\in\mathbb{N}$, of real numbers tending to $+\infty$, we divide both sides of (7.1) by $t=t_{i}$ and get that

[TABLE]

Thus the ground state $\mu$ is an $f$-maximizing measure.

Recall that by definition, the locking property implies in particular the uniqueness of maximizing measures. Thus each $f\in\operatorname{Lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})\,\cup\,\operatorname{lock}_{\sigma}(\Sigma^{+},d_{\bm{a}})$ admits a unique ground state.

Combining the above with Theorem 1.1 and Theorem 1.2, we get Theorem 1.3. ∎

Theorem 1.3 reveals a concrete connection between ergodic optimization and thermodynamical formalism through the maximizing measures and equilibrium states. In fact, there are deeper reasons for such a close relationship between these two fields. As more detailed discussions in [LZ23, Section 1] suggest, there is a “dictionary” for the correspondences between thermodynamical formalism and its tropical counterpart, ergodic optimization. In this dictionary, the Ruelle–Perron–Frobenius operators and their normalizations translate to the corresponding Bousch operators (a.k.a. Bousch–Lax operators) and their normalizations, the topological pressure (i.e., maximal free energy) to the maximal potential energy, the equilibrium states to the maximizing measures (measures of maximal potential energy), the existence and construction of the eigenfunction of the Ruelle–Perron–Frobenius operator (see for example, [Li18, Theorem 5.16]) to the existence and construction of the calibrated sub-action in [LZ23], and the existence, uniqueness, and equidistribution of periodic points for the equilibrium state may be considered as the counterparts to the existence, uniqueness, and periodicity of (the support of) the maximizing measure. In the current paper, we investigated in yet another “language”, namely, a probabilistic one through random maximum mean cycle problems on digraphs using probabilistic tools. Similar analogies can be drawn between these three “languages” in the “dictionary”. It would be interesting to investigate further into such connections, especially the probability aspects.