---
title: Ψ-Locally Constant Functions
url: https://www.emergentmind.com/topics/locally-constant-functions
type: topic
---

# Ψ-Locally Constant Functions

A Ψ-locally constant function is a mathematical construct that generalizes both slowly varying functions and additive locally constant functions by incorporating a controlling scale function Ψ or ψ. The concept arises in two distinct but related contexts: (1) in real and asymptotic analysis, where ψ-locally constant functions interpolate between classical slowly varying functions and locally constant functions, and (2) in symbolic dynamics, where "Ψ-locally constant" refers to vector-valued functions on subshifts of finite type that are constant on specified cylinders. Both variants have rigorous integral representations, closure properties, and a role in large deviation theory and thermodynamic formalism.

## 1. Definition and Integral Representation

A function $f : [x_0, \infty) \to (0, \infty)$ is ψ-locally constant (ψ-l.c.f.) if there exists a non-decreasing function $\psi(x) \ge 1$, with $\psi(x) = o(x)$ as $x \to \infty$, such that for any fixed $v \in \mathbb{R}$,
\[
\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.
\]
This condition interpolates between additive local constancy ($\psi(x) \equiv 1$) and slow variation ($\psi(x) = x$). For a suitable class $\mathcal{K}$ of scale functions ψ—namely, those that are non-decreasing, $\psi(x) = o(x)$, and satisfy a scale-stability property—every ψ-l.c.f. admits a Karamata-type integral representation:
\[
f(x) = c(x) \exp\left\{ \int_1^{\Psi(x)} \frac{\varepsilon(t)}{t} \, dt \right\}
\]
where $c(x) \to c \in (0, \infty)$, $\varepsilon(x) \to 0$ as $x \to \infty$, and $\Psi(x) := \int_1^x \psi(t)^{-1}\,dt$ is strictly increasing and divergent as $x \to \infty$ [1006.3164].

In symbolic dynamics, for a subshift of finite type $X \subset \mathcal{A}^{\mathbb{N}}$ defined by an adjacency matrix $A$, a function $\Phi \in C(X, \mathbb{R}^m)$ is called k-locally constant (often denoted LC$_k$) if it is constant on every cylinder of length $k$; that is, $\|\Phi\|_k = 0$ with
\[
\|\Phi\|_k = \sup\{ \|\Phi(x) - \Phi(y)\| : x_i = y_i \text{ for } i=1,\ldots,k \}.
\]
Such functions form the space LC$_k(X, \mathbb{R}^m) \cong \mathbb{R}^M$, where $M$ is the number of $k$-admissible blocks [1804.07822].

## 2. Properties and Closure Relations

The class of ψ-l.c.f. is closed under pointwise multiplication and inversion: if $f$ and $g$ are ψ-l.c.f., so are $f\,g$ and $1/f$ (as their logarithms satisfy the same increment property). This generalizes the closure properties of both slowly varying and locally constant functions. Furthermore, ψ-l.c.f. always exhibit subexponential growth in the Ψ-scale:
\[
\lim_{x \to \infty} \frac{\ln f(x)}{\Psi(x)} = 0.
\]
Classical slowly varying functions and locally constant functions are both recovered as special cases corresponding to choices $\psi(x) = x$ and $\psi(x) = 1$ respectively [1006.3164].

In dynamical systems, the k-locally constant functions (over a shift space $X$) comprise a finite-dimensional vector space whose structure can be exploited for an explicit topological classification of their thermodynamic and ergodic properties [1804.07822].

## 3. Classification and Cones of Potentials

For a subshift of finite type $X$ and a fixed order $k$, the Banach space LC$_k(X, \mathbb{R})$ can be partitioned into finitely many disjoint convex cones $U_1, \ldots, U_N$. Associated to each cone $U_i$ is a finite set of ergodic measures $M_i$ such that, for every potential $\varphi \in U_i$, the zero-temperature measure $\mu_{\infty, \varphi}$ is a convex combination of the measures in $M_i$. The three main types of cones are:
- Open and dense cones $U_i$ with $M_i$ a single k-elementary periodic orbit measure.
- Cones where all $\varphi$ are cohomologous to zero on a transitive subshift $X_i$, and $M_i$ is the unique measure of maximal entropy on $X_i$.
- Cones where $M_i$ is a set of measures of maximal entropy on several transitive components, and $\mu_{\infty,\varphi}$ is a convex combination with finitely many possible coefficient tuples [1804.07822].

This classification, originally developed for symbolic systems, applies with minor modifications to bidirectional or Markovian frameworks and is controlled by the geometry of the associated rotation sets.

## 4. Rotation Sets, Entropy Geometry, and Regularity

Given a potential $\Phi \in LC_k(X, \mathbb{R}^m)$, the rotation set $R(\Phi) = \{ rv(\mu) : \mu \in \mathcal{M}_\sigma \}$, where $rv(\mu) = (\int \phi_1\,d\mu, \ldots, \int \phi_m\,d\mu)$, is a convex polyhedron with vertices corresponding to rotation vectors of periodic orbit measures. Each face $F$ of $R(\Phi)$ corresponds to a supporting subsystem $X_F$, decomposing into finitely many transitive components.

The localized entropy function $\mathcal{H}(w) = \sup\{ h_\mu(\sigma): rv(\mu) = w \}$ is upper-semicontinuous, concave, and continuous on the interior of $R(\Phi)$, and piecewise $C^1$ on the boundary for $m=2$. Finer regularity (e.g., $C^2$ or analytic) does not generally hold, even on interiors of faces, as demonstrated by examples where the entropy drops or exhibits corners along certain faces [1804.07822].

In these settings, a universal potential $\Psi \in LC_k(X, \mathbb{R}^M)$ can encode all one-dimensional locally constant potentials: any $\varphi \in LC_k(X, \mathbb{R})$ can be written as $t \cdot (\alpha \cdot \Psi)$ for some $\alpha \in S^{M-1}$ and $t>0$, reducing zero-temperature analysis to the structure of the rotation polyhedron's boundary.

## 5. Illustrative Examples

A summary of key examples clarifies the ψ-l.c.f. and LC$_k$ frameworks:

| Setting                          | Function/Class                                      | Notes                                                                   |
|-----------------------------------|-----------------------------------------------------|-------------------------------------------------------------------------|
| $\psi(x) = 1$                    | Locally constant                                   | $f(x+v)/f(x)\to1$ for all $v$                                          |
| $\psi(x) = x$                    | Slowly varying                                     | $f(vx)/f(x)\to1$ for all $v$                                           |
| $\psi(x)=x^\alpha$, $0<\alpha<1$ | Nontrivial ψ-l.c.f.                                | $f(x) = \ln((1-\alpha)\ln x)$ is ψ-l.c.f.                              |
| $X=\{0,1\}^\mathbb{N}$, order $2$| $\varphi$ on 4-cylinders                           | Three open cones with periodic zero-temperature measure; Bernoulli, etc.|

- For the 2-symbol full shift of order 2, the possible zero-temperature measures for locally constant $\varphi$ are classified by the cone in parameter space: measures supported on periodic orbits, Bernoulli measures, or strictly convex combinations relating to Golden-Mean shifts. 
- For the 3-symbol full shift (order 2), only a small number of elementary periodic orbits appear generically as zero-temperature measures, even though the space of orbits is much larger [1804.07822].

In the ψ-l.c.f. context, ψ-l.c.f. functions are constructed that grow, for example, as $\ln((1-\alpha)\ln x)$, exemplifying behavior not captured by classical notions of regular variation [1006.3164].

## 6. Applications to Probability and Dynamical Systems

ψ-l.c.f. play a crucial role in broadening the conditions under which large deviation principles hold for sums of i.i.d. random variables, allowing replacement of regular variation with the weaker ψ-l.c.f. condition on tails. For example, for random variables with upper tails in a ψ-l.c.f. class, the principal large deviation asymptotics $\Pr\{S_n \ge x\} \sim n F_+(x)$ can be derived under appropriate growth conditions on $x$ and $n$ [1006.3164].

In dynamical systems, the geometric/topological classification of LC$_k$(X, ℝ)-potentials provides a complete description of zero-temperature limits and their measures, with zero-temperature measures explicitly determined by the faces of the rotation set polyhedron. All one-dimensional locally constant questions are reduced to examining these faces, grounding the thermodynamic formalism for shifts of finite type in polyhedral rotation geometry [1804.07822].

## 7. Computability, Generalizations, and Open Questions

Recent advances establish that rotation sets and their entropy functions are computable in principle from the data of the potential, enabling explicit determination of the cones $U_i$ and measures $M_i$. The geometric framework for LC$_k$ potentials avoids reliance on transfer-operator arguments and suggests possible extensions to non-uniformly hyperbolic systems, sub-additive potentials, and countable Markov shifts.

While the localized entropy function $\mathcal{H}$ is piecewise $C^1$ on the boundary for $m=2$, it need not be $C^2$ or analytic even within the relative interior of a face. The identification of possible "corners" and their dynamical significance remains an open area of research [1804.07822].

Source: https://www.emergentmind.com/topics/locally-constant-functions