---
title: 'Binary Shifts with a Hole: Dynamics & Dimensions'
url: https://www.emergentmind.com/topics/binary-shifts-with-a-hole
type: topic
---

# Binary Shifts with a Hole: Dynamics & Dimensions

Binary shifts with a hole are open symbolic dynamical systems on the one-sided full shift \(X=\{0,1\}^{\mathbb N}\) in which a prescribed set of itineraries is removed and one studies the sequences whose forward shifts never enter that set. In the lexicographic formulation, which is central to double-base expansions, the hole is the open interval \(]a,b[\) with \(a\in 0\{0,1\}^\infty\) and \(b\in 1\{0,1\}^\infty\), and the survivor subshift is
\[
\Omega_{a,b}:=\{i_1i_2\cdots\in\{0,1\}^\infty: i_ni_{n+1}\cdots\notin ]a,b[\ \text{for all }n\ge 1\}.
\]
In related settings, the hole is a cylinder \(H_w=[w]\) determined by a finite word \(w\), or an interval in a one-dimensional expanding map whose itineraries are coded by binary sequences. Across these formulations, the subject connects symbolic dynamics, Hausdorff dimension, thermodynamic formalism, \(\beta\)-expansions, open dynamical systems, and escape-rate theory [2509.04227].

## 1. Symbolic formulation and hole conventions

The common ambient space is the one-sided binary shift \(X=\{0,1\}^{\mathbb N}\) with shift \(\sigma:(i_1i_2\cdots)\mapsto(i_2i_3\cdots)\). A “hole” is a subset of \(X\), and the survivor set is the set of sequences whose forward iterates under \(\sigma\) never enter that hole. In the lexicographic framework, the hole is typically an open interval in symbol space; in the escape-rate framework, it is a cylinder determined by a finite forbidden prefix; in geometric realizations, it is an interval in phase space whose symbolic itineraries form a lexicographic exclusion region [2509.04227].

| Setting | Hole | Survivor set |
|---|---|---|
| Lexicographic symbolic model | \(]a,b[\subset\{0,1\}^{\mathbb N}\) | \(\Omega_{a,b}\) |
| Word-defined shift | \(H_w=[w]\) | \(S_n(H_w)\) and its asymptotics |
| Geometric interval map | \((A,B)\) or \([0,t)\subset[0,1]\) | Symbolically coded survivor subshift |

For the lexicographic model, the open interval is
\[
]a,b[:=\{u\in\{0,1\}^\infty: a\prec u\prec b\},
\]
and the associated survivor subshift is
\[
\Omega_{a,b}:=\{ i_1i_2\cdots\in\{0,1\}^\infty : i_ni_{n+1}\cdots \notin ]a,b[ \ \text{for all } n\ge 1 \}.
\]
This is a closed, shift-invariant set, hence a subshift. When the hole is defined by lexicographic constraints, the resulting survivor set is often a subshift of finite type or a sofic shift determined by forbidding the words associated to the hole.

A distinct but related convention fixes a finite word \(w=(w_0\cdots w_{r-1})\) and defines the hole by the cylinder
\[
H_w=[w]=\{x\in X:x_0\cdots x_{r-1}=w\}.
\]
The survival set after \(n\) iterates is
\[
S_n(H_w):=\{x\in X:\sigma^k(x)\notin H_w \text{ for } k=0,1,\dots,n\}.
\]
For the shift, this is equivalent to requiring that the first \(n+r\) symbols contain no occurrence of \(w\) as a contiguous sub-word.

A basic structural point is that “binary shift with a hole” is therefore not a single model but a family of closely related open systems. This suggests that statements about entropy, dimension, or escape rate depend strongly on the type of hole being used.

## 2. Entropy, Hausdorff dimension, and kneading equations

For lexicographic survivor subshifts, the central invariants admit explicit formulas. Given an admissible kneading pair \((a,b)\), define the projection map
\[
\pi_{q_0,q_1}(i_1i_2\cdots)=\sum_{k=1}^\infty \frac{i_k}{q_{i_1}q_{i_2}\cdots q_{i_k}},
\]
with \(q_{i_k}=q_0\) if \(i_k=0\) and \(q_{i_k}=q_1\) if \(i_k=1\). The kneading invariant is
\[
K_{a,b}(z_0,z_1)=\sum_{n=1}^\infty \big(b_n\,z_{b_1}\cdots z_{b_n}-a_n\,z_{a_1}\cdots z_{a_n}\big),
\]
so that \(K_{a,b}(q_0^{-1},q_1^{-1})=\pi_{q_0,q_1}(b)-\pi_{q_0,q_1}(a)\) [2509.04227].

The topological entropy of \(\Omega_{a,b}\) is
\[
h(\Omega_{a,b})=\log\beta,
\]
where \(\beta>1\) is the maximal solution of
\[
\pi_{\beta,\beta}(a)=\pi_{\beta,\beta}(b).
\]
For unequal contraction factors, the Hausdorff dimension of the projection is
\[
\dim_H \pi_{q_0,q_1}(\Omega_{a,b})=\min\{1,s\},
\]
where \(s>0\) is the maximal solution of
\[
\pi_{q_0^s,q_1^s}(a)=\pi_{q_0^s,q_1^s}(b).
\]
The same parameter \(s\) is characterized as the unique zero of the subadditive pressure-like relation
\[
\lim_{n\to\infty}\frac{1}{n}\log \sum_{i_1\cdots i_n\in L_{a,b,n}}\frac{1}{q_{i_1}^s\cdots q_{i_n}^s}=0,
\]
where \(L_{a,b,n}\) is the set of \(n\)-letter words appearing in \(\Omega_{a,b}\).

When \(\Omega_{a,b}\) is of finite type or sofic, one may build a finite labeled graph or automaton whose language coincides with \(\Omega_{a,b}\). If \(A\) is the adjacency matrix of this graph, then
\[
h_{\mathrm{top}}(\Omega_{a,b})=\log \rho(A),
\]
where \(\rho(A)\) is the spectral radius of \(A\). This gives a finite-state realization of the same entropy that appears through the kneading equation.

Two continuity statements are especially important. First, the dimension function
\[
d_{q_0,q_1}(a,b)=\dim_H \pi_{q_0,q_1}(\Omega_{a,b})
\]
is continuous in \((a,b)\). Second, the map
\[
(q_0,q_1)\mapsto \dim_H U_{q_0,q_1}
\]
is continuous for \(q_0,q_1>1\). In the equal-base case \(q_0=q_1=q\), the dimension formula reduces to
\[
\dim_H U_{q,q}=\frac{h(U_{q,q})}{\log q},
\]
which is the uniform-slope specialization of the general theory.

## 3. Double-base expansions and the univoque set

A principal source of binary shifts with a hole is the theory of double-base expansions. For \(q_0,q_1>1\), the projection
\[
\pi_{q_0,q_1}(i_1i_2\cdots)=\sum_{k=1}^\infty \frac{i_k}{q_{i_1}\cdots q_{i_k}}
\]
maps \(\{0,1\}^{\mathbb N}\) to the attractor
\[
J_{q_0,q_1}:=\pi_{q_0,q_1}(\{0,1\}^{\mathbb N}),
\]
which is the attractor of the iterated function system \(\{f_0(x)=x/q_0,\ f_1(x)=(x+1)/q_1\}\). The univoque set is
\[
U_{q_0,q_1}=\{x\in J_{q_0,q_1}:\#\,\pi_{q_0,q_1}^{-1}(x)=1\},
\]
the set of points having a unique \((q_0,q_1)\)-expansion [2509.04227].

For \(q_0,q_1>1\) with \(q_0+q_1\ge q_0q_1\), the unique-expansion set is characterized symbolically by two extremal kneading sequences:
- \(a_{q_0,q_1}\), the quasi-greedy expansion of \(1/q_1\), starting with \(01\);
- \(b_{q_0,q_1}\), the quasi-lazy expansion of \(1/(q_0(q_1-1))\), starting with \(10\).

Up to a countable set of endpoints,
\[
U_{q_0,q_1}=\{i_1i_2\cdots\in\{0,1\}^{\mathbb N}: i_ni_{n+1}\cdots\notin [a_{q_0,q_1},b_{q_0,q_1}] \ \text{for all } n\ge 1\},
\]
and the difference between \(\Omega_{a_{q_0,q_1},b_{q_0,q_1}}\) and \(U_{q_0,q_1}\) is countable. Consequently,
\[
\dim_H U_{q_0,q_1}=\dim_H \pi_{q_0,q_1}(\Omega_{a_{q_0,q_1},b_{q_0,q_1}}).
\]

The general dimension theorem yields several threshold regimes. The paper introduces generalized thresholds \(G(q_0)\) and \(K(q_0)\) and states:
- \(U_{q_0,q_1}\) is trivial for \(q_1<G(q_0)\);
- \(\dim_H U_{q_0,q_1}=0\) for \(q_1\le K(q_0)\);
- \(\dim_H U_{q_0,q_1}>0\) for \(q_1>K(q_0)\).

More precisely:
- If \(q_1\le K(q_0)\), then \(\dim_H U_{q_0,q_1}=0\).
- If \(K(q_0)<q_1<\frac{q_0}{q_0-1}\), then \(s\in(0,1)\) is the maximal root of
  \[
  \pi_{q_0^s,q_1^s}(\ell_{a_{q_0,q_1},b_{q_0,q_1}})=\pi_{q_0^s,q_1^s}(r_{a_{q_0,q_1},b_{q_0,q_1}}),
  \]
  and \(\dim_H U_{q_0,q_1}=s\).
- If \(q_1=\frac{q_0}{q_0-1}\), then \(\dim_H U_{q_0,q_1}=1\).
- If \(q_1>\frac{q_0}{q_0-1}\), then \(s\in(0,1)\) solves
  \[
  q_0^{-s}+q_1^{-s}=1,
  \]
  and \(\dim_H U_{q_0,q_1}=s\).

An explicit worked example is also given. Let \(q_0\approx 2.247\) be the real root of \(q_0^3=2q_0^2+q_0-1\), and let \(q_1=1+1/q_0\approx 1.445\). Then
\[
a_{q_0,q_1}=011(100)^\infty,\qquad b_{q_0,q_1}=(10)^\infty,
\]
the extremal elements are \(\ell=(011)^\infty\) and \(r=(10)^\infty\), the entropy satisfies
\[
h(\Omega_{(011)^\infty,(10)^\infty})=\log\beta,\qquad \beta^3=\beta+1,\qquad \beta\approx 1.325,
\]
and the Hausdorff dimension is the number \(s\) solving
\[
\frac{q_1^s+1}{q_0^s q_1^{2s}-1}=\frac{q_0^s}{q_0^s q_1^s-1},
\]
giving \(s\approx 0.512255\).

## 4. Doubling maps, linear Lorenz maps, and greedy systems with a hole

The lexicographic survivor subshift \(\Omega_{a,b}\) has direct realizations in one-dimensional dynamics. For the doubling map
\[
T(x)=2x \bmod 1
\]
with a hole \(H=(A,B)\subset[0,1]\), binary coding identifies the geometric survivor set with \(\pi_{2,2}(\Omega_{a,b})\), where \(a\) and \(b\) are determined by the binary expansions of \(A\) and \(B\). Its entropy is
\[
h(\Omega_{a,b})=\log\beta \quad \text{where } \pi_{\beta,\beta}(a)=\pi_{\beta,\beta}(b),
\]
and because the doubling map has uniform slope \(2\),
\[
\dim_H(\text{survivor set})=\frac{h_{\mathrm{top}}(\Omega_{a,b})}{\log 2}.
\]
For the central hole \(H=(1/3,2/3)\), the endpoints have binary expansions \((01)^\infty\) and \((10)^\infty\), the survivor subshift is \(\Omega_{(01)^\infty,(10)^\infty}\), and
\[
h(\Omega_{(01)^\infty,(10)^\infty})=0,
\qquad
\dim_H(\text{survivor})=0
\]
[2509.04227].

Linear Lorenz maps fit the same framework. Their left and right branches have constant slopes \(q_0,q_1>1\), the discontinuity itineraries define kneading sequences \(a,b\), and the symbolic hole is again \(]a,b[\). The resulting entropy and Hausdorff dimension are
\[
h(\Omega_{a,b})=\log\beta \quad \text{where } \pi_{\beta,\beta}(a)=\pi_{\beta,\beta}(b),
\]
and
\[
\dim_H \pi_{q_0,q_1}(\Omega_{a,b})=\min\{1,s\},\qquad
\pi_{q_0^s,q_1^s}(a)=\pi_{q_0^s,q_1^s}(b).
\]

A further realization comes from intermediate \(\beta\)-transformations. For \((\beta,\alpha)\in\Delta=\{(b,a):b\in(1,2),\ a\in[0,2-b]\}\), the piecewise linear map
\[
T_{\beta,\alpha}(x)=
\begin{cases}
\beta x+\alpha,& x<p,\\
\beta x+\alpha-1,& x\ge p,
\end{cases}
\qquad
p=\frac{1-\alpha}{\beta},
\]
has upper and lower symbolic codings \(\tau^\pm_{\beta,\alpha}\) and associated subshifts \(\Omega^\pm_{\beta,\alpha}\). The central conjugacy result states that for every \((\beta,\alpha)\in\Delta\), there exist \(t\in[0,1]\) and \(\beta'\in(1,2)\) such that
\[
(T_{\beta,\alpha},[0,1])\ \text{is topologically conjugate to}\ \big(T_{\beta',0}\big|_{K^+_{\beta',0}(t)},\,K^+_{\beta',0}(t)\big),
\]
where
\[
K^+_{\beta,\alpha}(t)=\{x\in[0,1):T_{\beta,\alpha}^n(x)\notin [0,t)\ \text{for all }n\ge 0\}
\]
is the survivor set for a hole at zero [2211.14584].

The correspondence is not one-to-one. The paper constructs a value of \(\beta\) with minimal polynomial \(x^5-x^4-x^3-2x^2+x+1\) and a sequence \(\xi=00(011)^\infty\) showing that some greedy-with-hole systems do not arise from an intermediate system. This is a precise obstruction, not merely a technical artifact.

For the binary map, the hole-at-zero formalism becomes particularly concrete. If \(H=[0,1/4)\), then the kneading word is
\[
s=\tau^+_{2,0}(1/4)=0100^\infty,
\]
and the condition \(\sigma^n(\omega)\succeq 0100^\infty\) for all \(n\) is equivalent to forbidding the word \(00\). The survivor subshift is therefore the SFT with adjacency matrix
\[
A_H=\begin{pmatrix}
0&1\\
1&1
\end{pmatrix},
\qquad
\rho(A_H)=\frac{1+\sqrt5}{2}=\varphi,
\]
so
\[
h_{\mathrm{top}}=\log\varphi,
\qquad
\dim_H\big(K^+_{2,0}(1/4)\big)=\frac{\log\varphi}{\log 2}\approx 0.6942.
\]

## 5. \(\beta\)-transformations with interval holes and devil’s staircase structure

For \(\beta\in(1,2)\), the map
\[
T_\beta(x)=\beta x \pmod 1
\]
generates a binary symbolic coding constrained by Parry’s quasi-greedy expansion \(d_\beta(1)\). The associated \(\beta\)-shift is
\[
X_\beta=\{(x_i)_{i\ge 1}:\sigma^j((x_i))\preceq d_\beta(1)\ \text{for all }j\}.
\]
Given an interval hole \((a,b)\subset(0,1)\), the survivor set is
\[
J_\beta(a,b)=\{x\in(0,1):T_\beta^n(x)\notin(a,b)\ \text{for all }n\ge 0\},
\]
and symbolically it is the set of sequences in \(X_\beta\) whose every shift avoids the lexicographic interval between the greedy expansions of \(a\) and \(b\) [1412.6384].

The paper studies three parameter sets:
\[
D_0(\beta)=\{(a,b)\in[0,1)^2:J_\beta(a,b)\neq\emptyset\},
\]
\[
D_1(\beta)=\{(a,b)\in[0,1)^2:J_\beta(a,b)\ \text{is uncountable}\},
\]
\[
D_2(\beta)=\{(a,b)\in[0,1)^2:B_\beta(a,b)\ \text{is finite}\},
\]
where \(B_\beta(a,b)\) is the set of bad periods, meaning those \(n\) for which every \(n\)-cycle intersects the hole. The inclusions
\[
D_2(\beta)\subset D_1(\beta)\subset D_0(\beta)
\]
hold.

A central organizing mechanism is the combinatorics of balanced words, Sturmian limits, and maximal extremal pairs. Rational slopes \(\gamma=p/q\) determine cyclically balanced words \(w_\gamma\), and from these one forms pairs \((s_\gamma,t_\gamma)\) by taking the lexicographically largest cyclic permutation starting with \(0\) and the lexicographically smallest cyclic permutation starting with \(1\). The function \(\beta\mapsto \gamma(\beta)\), which selects the maximal admissible balanced word, is a devil’s staircase: it is continuous, non-decreasing, and has derivative zero almost everywhere.

Inside the nontrivial region
\[
I_\beta=(0,d_\beta(1))\times(1/\beta,1),
\]
the boundaries of \(D_0(\beta)\), \(D_1(\beta)\), and \(D_2(\beta)\) are described by maximal extremal pairs and their Farey descendants. For maximal pairs, the survivor set at the boundary hole is essentially a single periodic orbit and its allowable preimages; for descendant pairs it is countable; after shrinking the hole away from the boundary by \(\varepsilon>0\), one obtains uncountable survivor sets and, under the coprime-length condition \(\gcd(j,q)=1\), only finitely many bad periods.

The paper also identifies an “unbalanced” region
\[
R_\beta=(0,0^n1)\times(d_\beta(1),0^{n-1}1),
\]
when \(d_\beta(1)\in[1/(n+1),1/n)\). In this regime, balanced words alone do not produce all maximal extremal pairs. Additional pairs are built from a Farey-like tree with roots \(0\) and a suitable minimal cyclic permutation \(u_\gamma\). This extends the doubling-map picture to general \(\beta\in(1,2)\) and introduces a genuine admissibility phenomenon coming from the Parry constraint \(d_\beta(1)\).

## 6. Escape rates, maximal holes, and ordering phenomena

When the hole is a cylinder \(H_w=[w]\) specified by a finite word \(w\), the relevant invariant is the escape rate. For Bernoulli measure \(\mu\), let
\[
p_n=\mu(S_n(H_w)).
\]
The escape rates are
\[
\gamma_H^-=\liminf_{n\to\infty} -\frac1n\log p_n,
\qquad
\gamma_H^+=\limsup_{n\to\infty} -\frac1n\log p_n,
\]
and the paper focuses on \(\gamma_H=\gamma_H^-\). The weighted autocorrelation polynomial \(c_w\) and the polynomial
\[
\tau_w(z):=\mu(w)z^r+(1-z)c_w(p_{a_1},\dots,p_{a_A},z)
\]
determine the escape rate by
\[
\gamma_{H_w}=\log z_0,
\]
where \(z_0>1\) is the smallest positive root of \(\tau_w\) [2112.14248].

For a prime hole, meaning a word whose autocorrelation vector is \((1,0,\dots,0)\), the formula simplifies to
\[
\tau_w(z)=\mu(w)z^r-z+1.
\]
For an all-one-symbol hole \(w=a^r\),
\[
\tau_w(z)=\frac{p_a^r(1-p_a)z^{r+1}-z+1}{1-p_az}.
\]

The maximization problem is especially explicit in the binary Bernoulli case. Let \(p\) be the larger symbol probability and \(q\) the smaller, and fix a length \(r\ge 2\). There are two distinguished candidates:
- \(P^r\): prime holes of length \(r\) with maximal prime measure, such as \(a^{r-1}b\), of measure \(p^{r-1}q\);
- \(M^r\): holes of maximal measure, namely \(a^r\), of measure \(p^r\).

The maximal escape rate over all holes of length \(r\) is achieved either by a prime hole in \(P^r\) or by a maximal-measure hole in \(M^r\). In the binary case,
\[
\gamma_{\max}^r=
\begin{cases}
\gamma_{w_P},& p\in[1/2,\,1-1/(r+1)],\\
\gamma_{w_M},& p\in[1-1/(r+1),\,1).
\end{cases}
\]
Moreover, on the subinterval \(p\in[1-1/r,\,1-1/(r+1)]\),
\[
\gamma_{w_P}=\log(1/p).
\]

Two comparison principles follow. First, if two holes have the same length and the same measure, then a prime hole has larger escape rate than a non-prime hole. Second, for prime holes of the same length, larger measure implies larger escape rate. These statements give a clean classification of maximizers, but they do not support a universal ordering principle.

Indeed, a major negative result is that, for non-equiprobable symbols, ordering holes by escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. The paper exhibits explicit counterexamples. This sharply distinguishes the non-equiprobable case from the equiprobable case, where classical periodic-orbit ordering phenomena do hold.

The Markov-measure case is more intricate. The escape polynomial acquires additional factors involving
\[
\chi_\Pi=\pi_{aa}+\pi_{bb}-1,
\]
and the sign of \(\chi_\Pi\) affects comparisons. For \(\chi_\Pi>0\), prime holes remain preferable in the same-measure comparison; for \(\chi_\Pi<0\), non-prime holes can dominate in certain regions. A plausible implication is that “binary shifts with a hole” support several distinct extremal theories, depending not only on the hole but also on the background invariant measure.

## 7. Structural themes and scope

Several themes recur across the literature. One is the equivalence between symbolic survivor sets and geometric survivor sets: the same lexicographic exclusion mechanism appears in univoque expansions, the doubling map with a hole, linear Lorenz maps, greedy and intermediate \(\beta\)-transformations, and the \(\beta\)-transformation with a general interval hole [2509.04227].

A second theme is the coexistence of exact formulas and highly singular parameter dependence. Continuity of entropy and Hausdorff dimension coexists with devil’s staircase behavior in parameter selections such as \(\beta\mapsto\gamma(\beta)\) and the dimension function \(\eta_{\beta,\alpha}(t)=\dim_H(K^+_{\beta,\alpha}(t))\), which is decreasing and constant Lebesgue almost everywhere. The bifurcation set
\[
E^+_{\beta,\alpha}=\{t\in[0,1):T_{\beta,\alpha}^n(t)\notin[0,t)\ \text{for all }n\ge 0\}
\]
is Lebesgue null but has full Hausdorff dimension \(1\) [2211.14584].

A third theme is the role of finite-type and sofic approximations. When kneading invariants are periodic or eventually periodic, finite automata and adjacency matrices give explicit computations of entropy and dimension. For non-finite-type systems, the same quantities are recovered through root equations, pressure, and limiting approximations.

Taken together, these results position binary shifts with a hole as a unifying framework for open symbolic dynamics. The framework is simultaneously combinatorial, metric, and geometric: holes are specified by forbidden words or forbidden lexicographic intervals; survivor sets are subshifts, attractors, or repellers; and the principal invariants are entropy, Hausdorff dimension, and escape rate. The resulting theory extends from the full two-shift to constrained \(\beta\)-shifts, from uniform to non-uniform contractions, and from deterministic symbolic exclusions to measure-dependent leakage.

Source: https://www.emergentmind.com/topics/binary-shifts-with-a-hole