Ψ-Locally Constant Functions

• Ψ-locally constant functions are defined by a scaling function Ψ that interpolates between additive local constancy and slow variation, ensuring the ratio f(x+vψ(x))/f(x) converges to 1.
• Their Karamata-type integral representation and closure under multiplication and inversion provide robust tools for large deviation theory and subexponential growth analysis.
• In symbolic dynamics, LCₖ functions classify potentials over subshifts, enabling explicit topological and thermodynamic analyses through rotation set geometry and convex cone decomposition.

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: $\psi(x) \ge 1$0 where $\psi(x) \ge 1$1, $\psi(x) \ge 1$2 as $\psi(x) \ge 1$3, and $\psi(x) \ge 1$4 is strictly increasing and divergent as $\psi(x) \ge 1$5 (Borovkov et al., 2010).

In symbolic dynamics, for a subshift of finite type $\psi(x) \ge 1$6 defined by an adjacency matrix $\psi(x) \ge 1$7, a function $\psi(x) \ge 1$8 is called k-locally constant (often denoted LC$\psi(x) \ge 1$9) if it is constant on every cylinder of length $\psi(x) = o(x)$0; that is, $\psi(x) = o(x)$1 with

$\psi(x) = o(x)$2

Such functions form the space LC$\psi(x) = o(x)$3, where $\psi(x) = o(x)$4 is the number of $\psi(x) = o(x)$5-admissible blocks (Wolf et al., 2018).

2. Properties and Closure Relations

The class of ψ-l.c.f. is closed under pointwise multiplication and inversion: if $\psi(x) = o(x)$6 and $\psi(x) = o(x)$7 are ψ-l.c.f., so are $\psi(x) = o(x)$8 and $\psi(x) = o(x)$9 (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: $x \to \infty$0 Classical slowly varying functions and locally constant functions are both recovered as special cases corresponding to choices $x \to \infty$1 and $x \to \infty$2 respectively (Borovkov et al., 2010).

In dynamical systems, the k-locally constant functions (over a shift space $x \to \infty$3) comprise a finite-dimensional vector space whose structure can be exploited for an explicit topological classification of their thermodynamic and ergodic properties (Wolf et al., 2018).

3. Classification and Cones of Potentials

For a subshift of finite type $x \to \infty$4 and a fixed order $x \to \infty$5, the Banach space LC$x \to \infty$6 can be partitioned into finitely many disjoint convex cones $x \to \infty$7. Associated to each cone $x \to \infty$8 is a finite set of ergodic measures $x \to \infty$9 such that, for every potential $v \in \mathbb{R}$0, the zero-temperature measure $v \in \mathbb{R}$1 is a convex combination of the measures in $v \in \mathbb{R}$2. The three main types of cones are:

• Open and dense cones $v \in \mathbb{R}$3 with $v \in \mathbb{R}$4 a single k-elementary periodic orbit measure.
• Cones where all $v \in \mathbb{R}$5 are cohomologous to zero on a transitive subshift $v \in \mathbb{R}$6, and $v \in \mathbb{R}$7 is the unique measure of maximal entropy on $v \in \mathbb{R}$8.
• Cones where $v \in \mathbb{R}$9 is a set of measures of maximal entropy on several transitive components, and $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$0 is a convex combination with finitely many possible coefficient tuples (Wolf et al., 2018).

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 $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$1, the rotation set $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$2, where $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$3, is a convex polyhedron with vertices corresponding to rotation vectors of periodic orbit measures. Each face $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$4 of $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$5 corresponds to a supporting subsystem $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$6, decomposing into finitely many transitive components.

The localized entropy function $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$7 is upper-semicontinuous, concave, and continuous on the interior of $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$8, and piecewise $$\lim_{x \to \infty} \frac{f(x + v\,\psi(x))}{f(x)} = 1.$$9 on the boundary for $\psi(x) \equiv 1$0. Finer regularity (e.g., $\psi(x) \equiv 1$1 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 (Wolf et al., 2018).

In these settings, a universal potential $\psi(x) \equiv 1$2 can encode all one-dimensional locally constant potentials: any $\psi(x) \equiv 1$3 can be written as $\psi(x) \equiv 1$4 for some $\psi(x) \equiv 1$5 and $\psi(x) \equiv 1$6, 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$\psi(x) \equiv 1$7 frameworks:

Setting	Function/Class	Notes
$\psi(x) \equiv 1$8	Locally constant	$\psi(x) \equiv 1$9 for all $\psi(x) = x$0
$\psi(x) = x$1	Slowly varying	$\psi(x) = x$2 for all $\psi(x) = x$3
$\psi(x) = x$4, $\psi(x) = x$5	Nontrivial ψ-l.c.f.	$\psi(x) = x$6 is ψ-l.c.f.
$\psi(x) = x$7, order $\psi(x) = x$8	$\psi(x) = x$9 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 $\mathcal{K}$0 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 (Wolf et al., 2018).

In the ψ-l.c.f. context, ψ-l.c.f. functions are constructed that grow, for example, as $\mathcal{K}$1, exemplifying behavior not captured by classical notions of regular variation (Borovkov et al., 2010).

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 $\mathcal{K}$2 can be derived under appropriate growth conditions on $\mathcal{K}$3 and $\mathcal{K}$4 (Borovkov et al., 2010).

In dynamical systems, the geometric/topological classification of LC$\mathcal{K}$5(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 (Wolf et al., 2018).

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 $\mathcal{K}$6 and measures $\mathcal{K}$7. The geometric framework for LC$\mathcal{K}$8 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{K}$9 is piecewise $\psi(x) = o(x)$0 on the boundary for $\psi(x) = o(x)$1, it need not be $\psi(x) = o(x)$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 (Wolf et al., 2018).