β-Transformation Dynamical System

• β-transformation-driven dynamical systems are non-integer base expansion models with chaotic, ergodic, and spectral properties.
• They utilize a piecewise linear map to produce greedy expansions, yielding explicit invariant densities and effective transfer operator analyses.
• Extensions to random and open systems reveal complex symbolic structures, quantifiable decay rates, and fractal survivor sets.

A $\beta$-transformation-driven dynamical system is a paradigmatic example of a non-integer base expansion system, central to ergodic theory, symbolic dynamics, and the spectral analysis of transfer operators. The deterministic and random $\beta$-transformations model a range of phenomena from number-theoretic expansions to chaotic statistical properties and explicit computation of invariant measures. Especially in the quadratic Parry case or under randomization, transfer operator asymptotics and symbolic structures show sharp spectral and ergodic phenomena.

1. Definition and Structure of the $\beta$-Transformation System

Let $\beta > 1$ be real. The classical $\beta$-transformation is defined by

$$T_\beta : [0,1) \rightarrow [0,1),\quad T_\beta(x) = \beta x - \lfloor \beta x \rfloor.$$

This map is piecewise linear with uniform slope $\beta$, expanding on a Markov partition into fundamental intervals $I_k = [k/\beta, (k+1)/\beta)$, $k = 0,\dots,\lfloor\beta\rfloor$. $T_\beta$ generates the greedy $\beta$-expansion of $x$: $$x = \sum_{k=1}^\infty x_k \beta^{-k},\quad x_k = \lfloor \beta T_\beta^{k-1}(x) \rfloor.$$ For particular algebraic $\beta$ (e.g., quadratic Parry numbers, satisfying $\beta^2 = a_0\beta + a_1$ for $a_0 \ge a_1 \ge 1$), every $x\in[0,1)$ admits an eventually periodic $\beta$-expansion, generalizing the theory of shift spaces.

The system extends naturally to random $\beta$-transformations by allowing $\beta$ to vary according to some stochastic process, resulting in a skew-product

$$R:(\omega,x) \mapsto (\theta \omega, T_{\beta(\omega)}(x)),$$

where $\theta$ is ergodic and $\beta(\omega)$ is a random variable, producing i.i.d. or non-i.i.d. random systems (Suzuki, 2023).

2. Transfer Operator and Spectral Theory

The Perron–Frobenius (transfer) operator $\mathscr{P}$ for $T_\beta$ is defined weakly via

$$\int_0^1 (\mathscr{P}f)(x)g(x)\,dx = \int_0^1 f(x)g(T_\beta(x))\,dx.$$

For $\beta$ in the quadratic Parry case ($\beta^2 = a_0\beta + a_1$), $T_\beta$ has $a_0+1$ inverse branches, yielding: $$(\mathscr{P}f)(x) = \frac{1}{\beta} \sum_{j=0}^{a_0-1} f\left(\frac{x+j}{\beta}\right) + \frac{1}{\beta} f\left(\frac{x+a_0}{\beta}\right) \chi_{[0,\,a_1/\beta)}(x)$$ (Cornean et al., 24 Feb 2025). In the integer case ($\beta \in \mathbb{N}$), this reduces to the averaging operator.

$\mathscr{P}$ has a unique invariant density $u$ ($\mathscr{P}u=u$). In the quadratic Parry case, $u$ is explicit and piecewise-affine. The second-largest eigenvalue and eigenfunctions, especially at $\lambda = \beta^{-1}$, dictate relaxation rates. For smooth $F$ with $\int F = 1$,

$$\mathscr{P}^k F(x) = u(x) + \beta^{-k}[F(1)-F(0)]v(x) + o(\beta^{-k}),$$

where $v$ is the explicitly-constructed eigenfunction with $\mathscr{P}v = \beta^{-1} v$ and $\int_0^1 v=0$ (Cornean et al., 24 Feb 2025). In the integer case, a complete asymptotic expansion using Bernoulli polynomials is available, but for non-integer $\beta$ only a two-term expansion exists due to the operator's continuous spectrum.

3. Random Beta-Systems and Invariant Measures

In the random setting, for sequences of randomly chosen $\beta(\omega)$ (i.i.d. or more generally ergodic), with a mean-expansion condition such as

$\int \frac{1}{\beta(\omega)}\,dP(\omega) < 1,$

one establishes a unique absolutely continuous invariant measure (acim) for the skew product. In the i.i.d. case, the invariant density $h$ is given by (Suzuki, 2023): $$h(x) = \frac{\phi(x)}{\int_0^1 \phi(y)\,dy},\quad \phi(x)=1_{[0,1]}(x) + \sum_{n=1}^\infty \int_{\Omega^n} \frac{1_{[0,T_{\omega_1\cdots\omega_n}(1)]}(x)}{\prod_{i=1}^n \beta(\omega_i)}\,dP^n(\omega_1,\dots,\omega_n).$$ In the Bernoulli finite-map setting, $h(x)$ and its derivatives with respect to probabilities depend analytically (linear response) on the randomization.

For non-i.i.d. or perturbed deterministic settings (e.g., strongly expanding or small perturbations of a non-simple base), existence and uniqueness results are deduced via operator invertibility and explicit series representations (Suzuki, 2023).

4. Dynamical and Symbolic Properties

The orbit structure of $T_\beta$ is determined by admissibility of greedy expansions, governed by the lexicographical constraint relative to the expansion of 1. The $\beta$-shift $X_\beta$ is the set of all allowed sequences, and $T_\beta$ is isomorphic to the left shift restricted to $X_\beta$ (Glazunov, 2011).

For $\beta$ of integer value, $X_\beta$ is the full shift; for non-integer Pisot numbers or Parry numbers, $X_\beta$ often has sofic or finite-type properties. Periodic points of $T_\beta$ are precisely points with eventually periodic greedy expansions.

Coding and combinatorial complexity are dictated by $\beta$. The number of admissible $n$-blocks in $X_\beta$ grows asymptotically like $\beta^n$ (exponentially), with polynomial deviation in certain Pisot cases. This structure forms the basis for symbolic bifurcations and for studying open systems with holes, escape rates, and dimension theory.

5. Open Systems, Survivor Sets, and Holes

For $\beta$-transformations with a hole, i.e., removal of an interval $[0,t)$, the survivor set

$$K_\beta(t) = \{x \in [0,1): T_\beta^n(x) \notin [0,t),\forall\, n\ge 0\}$$

defines the points whose orbits never visit the hole.

The Hausdorff dimension function $t \mapsto \dim_H K_\beta(t)$ is a Devil's staircase: non-increasing, constant on many intervals, with discontinuities at a Cantor-like set. The critical value $\tau(\beta)$, marking the threshold above which no survivor set has positive dimension, has a canonical symbolic description via extended Farey words and substitutions, and is given, in almost all intervals, by explicit periodic expansions (Allaart et al., 5 Nov 2024, Allaart et al., 2021). The structure of $K_\beta(t)$, the bifurcation set, and the corresponding dimension phenomena admit full combinatorial classification.

For open systems with holes not at zero, or with more general holes, similar combinatorial techniques using extremal pairs of balanced words and Farey descendants characterize the minimal condition for the existence and cardinality (countable, uncountable) of the survivor sets, as well as the set of "bad periods" (periods with no surviving orbits) (Clark, 2014).

6. Extensions: Random Walks, Alternate Bases, and Sierpiński Dynamics

Random-walk adic extensions of $\beta$-transformations involve skew-products with group-valued cocycles. These systems are infinite-measure-preserving, conservative, ergodic, and exhibit distributional stability and bounded-rational ergodicity, as established via asymptotics of local-limit theorems and explicit combinatorics of cylinder-sets (Bromberg, 2015).

Alternate-base greedy and lazy $\boldsymbol{\beta}$-transformations, where the digit expansion alternates between multiple bases $(\beta_0,\dots,\beta_{p-1})$, admit unique absolutely continuous invariant measures relative to the $p$-fold Lebesgue measure, are ergodic, and have metric entropy $\frac{1}{p}\log(\prod \beta_i)$. Frequency formulas for digits and explicit isomorphism constructions connect these systems to standard $\beta$-shifts (Charlier et al., 2021).

Dynamical systems on Sierpiński gaskets driven by $\beta$-transformations (for $1<\beta<2$) exhibit rich random, greedy, and lazy expansion behaviors in higher dimensions, unique maximal-entropy measures, and explicit phase transitions in topological structure as $\beta$ varies (Zhang et al., 2022).

7. Applications and Asymptotics

Sharp asymptotics for $\mathscr{P}^k F$ provide quantitative decay rates for correlations, determine statistical properties of $\beta$-expansions, and establish the precise role of boundary data (e.g., $F(1)-F(0)$) in first-order corrections for convergence to equilibrium (Cornean et al., 24 Feb 2025).

Absolutely continuous measures for random $\beta$-systems determine explicit digit statistics and error distributions in randomized expansions, with analytic dependence on parameters in the Bernoulli case, and explicit linear response (Suzuki, 2023).

The Devil's staircase and symbolic combinatorics in open $\beta$-systems connect with entropy plateaus, dimension theory, and bifurcation analysis. These results link the fine-scale symbolic and fractal structure to ergodic-theoretic and measure-theoretic phenomena.

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Table: Invariant Measures in Deterministic and Random $\beta$-Systems

System Type	Invariant Density (acim)	Source
Deterministic	$\mu_\beta(dx) = u(x) dx$, $u$ explicit (Parry), piecewise	(Cornean et al., 24 Feb 2025, Glazunov, 2011)
Random i.i.d.	$$h(x) \propto 1+\sum_{n\ge1} \int \frac{1_{[0,T^n(1)]}(x)}{\prod \beta_i}\,dP$$	(Suzuki, 2023)
Random Non-i.i.d.	$h$ via series with coefficients solving $(I+S)c=0$	(Suzuki, 2023)
Quadratic Parry	Piecewise-affine $u$, eigenfunctions $v$	(Cornean et al., 24 Feb 2025)
Alternate Base	$\mu_{\boldsymbol{\beta}}$ on $X_p$, density constant in each fiber	(Charlier et al., 2021)

The explicit construction of invariant densities underlies analysis of ergodic properties, statistical asymptotics, and the spectral profile of transfer operators.

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References:

• (Cornean et al., 24 Feb 2025): Sharp iteration asymptotics for transfer operators induced by greedy $\beta$-expansions
• (Suzuki, 2023): Absolutely continuous invariant measures for random dynamical systems of beta-transformations
• (Allaart et al., 2021, Allaart et al., 5 Nov 2024): Critical values and full theory for the $\beta$-transformation with a hole
• (Glazunov, 2011): Transformations, Dynamics and Complexity
• (Bromberg, 2015): Ergodic Properties of the Random Walk Adic Transformation over the Beta Transformation
• (Charlier et al., 2021): Dynamical behavior of alternate base expansions
• (Zhang et al., 2022): Random $\beta$-transformation on fat Sierpiński gasket