Martin's Conjecture: Invariant Operations on Turing Degrees

• Martin's Conjecture is a framework that classifies Turing-invariant functions on Cantor space into those that are eventually trivial or eventually increasing, forming a sharp dichotomy on a cone.
• It connects determinacy, Turing reducibility, and descriptive set theory by analyzing uniform, order-preserving, and measure-preserving functions through concrete classification theorems.
• Recent advances refine the theory using local analysis and Wadge-theoretic methods, revealing natural jump-like operators and prompting new questions in analogous degree structures.

Martin’s conjecture is a determinacy-theoretic classification program for degree-invariant operations on the Turing degrees. In its classical form, it concerns Turing-invariant functions $f:2^\omega\to 2^\omega$, viewed modulo eventual agreement on a Turing cone, and predicts a sharp dichotomy: such a function is either eventually trivial or eventually increasing in Turing degree; moreover, the increasing functions are organized by iterates of the Turing jump in a prewellordered hierarchy (Marks et al., 2011).

1. Classical formulation

Let $2^\omega$ be Cantor space. For reals $x,y\in 2^\omega$, write $x\le_T y$ for Turing reducibility and $x\equiv_T y$ for mutual Turing reducibility. A Turing-invariant function is a map $f:2^\omega\to 2^\omega$ such that

$x\equiv_T y \implies f(x)\equiv_T f(y).$

Such a function factors through the quotient of $2^\omega$ by $\equiv_T$, i.e. through the Turing degrees $\mathcal D$ (Bard, 2019).

The relevant notion of largeness is a Turing cone: for a real $2^\omega$0, the cone above $2^\omega$1 is

$2^\omega$2

Under determinacy, one studies functions only up to behavior on a cone. In the measure-theoretic presentation, Martin’s cone theorem yields a $2^\omega$3-$2^\omega$4 valued “Martin measure” on Turing-invariant sets, and statements “almost everywhere” mean “on a cone” (Marks et al., 2011).

For Turing-invariant functions $2^\omega$5, the Martin preorder is

$2^\omega$6

In the formulation used in the 2019 local analysis, Martin’s conjecture in $2^\omega$7 has two parts (Bard, 2019):

1. If $2^\omega$8 is Turing invariant, then either $2^\omega$9 on a cone, or there exists $x,y\in 2^\omega$0 such that $x,y\in 2^\omega$1 on a cone.
2. The Turing-invariant functions $x,y\in 2^\omega$2 with $x,y\in 2^\omega$3 are pre-well-ordered by $x,y\in 2^\omega$4, and if $x,y\in 2^\omega$5 has $x,y\in 2^\omega$6-rank $x,y\in 2^\omega$7, then $x,y\in 2^\omega$8 has rank $x,y\in 2^\omega$9.

The first part is the eventual-triviality/eventual-increase dichotomy; the second asserts that the nontrivial part of the hierarchy is governed by the jump. In expository treatments this is often summarized by saying that, under determinacy, the only natural Turing-degree-invariant operations are constants and jump-like operators (Marks et al., 2011).

2. Uniform versions and the local theorem

A central refinement is the uniform Martin conjecture, which restricts attention to uniformly Turing-invariant functions. Uniformity means that witnesses to $x\le_T y$0 or $x\le_T y$1 between inputs can be transformed effectively into witnesses for the corresponding relation between outputs. In the notation of the local paper, $x\le_T y$2 is uniformly Turing invariant (UTI) if there is a function $x\le_T y$3 such that

$x\le_T y$4

and uniformly order-preserving (UOP) if there is $x\le_T y$5 translating reductions $x\le_T y$6 into reductions $x\le_T y$7 uniformly (Bard, 2019).

For this uniform setting, Steel proved Part II, and Slaman–Steel proved Part I under $x\le_T y$8. The 2019 paper “Uniform Martin’s conjecture, locally” isolates a single-degree theorem: $x\le_T y$9 Equivalently, if a non-constant uniformly invariant function is defined on one Turing degree $x\equiv_T y$0, then its value lies in a degree $x\equiv_T y$1 with $x\equiv_T y$2 (Bard, 2019).

This local statement globalizes via Turing determinacy (TD). If $x\equiv_T y$3 is UTI on a cone, then either $x\equiv_T y$4 on a cone, or $x\equiv_T y$5 is literally constant on a cone. The same paper shows that, over $x\equiv_T y$6, this UTI-on-a-cone form of Part I is equivalent to TD. In particular, Part I of the uniform Martin conjecture is calibrated exactly by Turing determinacy rather than by full $x\equiv_T y$7 (Bard, 2019).

The local perspective also reframes Part II. Earlier global work of Becker, and later Kihara–Montalbán, showed that under $x\equiv_T y$8, UTI functions above the identity are, on a cone, Turing equivalent to pointclass jump operators, with the hierarchy tied to Wadge-theoretic structure. The 2019 paper asks whether an analogous classification exists degree-by-degree: for a fixed degree $x\equiv_T y$9, are all UTI functions $f:2^\omega\to 2^\omega$0 generated from constants, identity, complement, and local pointclass jumps by natural closure operations? That question remains open (Bard, 2019).

3. Proven cases on the Turing degrees

A substantial part of the modern progress on Martin’s conjecture concerns large subclasses of Turing-invariant functions for which Part I, and in some cases Part II, are known.

The uniform case is the oldest major success. Part I holds for uniformly Turing-invariant functions, and Part II holds for uniformly Turing-invariant functions as well. Becker’s characterization identifies strictly increasing uniformly invariant functions with universal sets for reasonable pointclasses; in particular, pointclass jump operators provide the canonical nontrivial examples (Marks et al., 2011).

A different line of attack concerns order-preserving functions. A Turing-invariant $f:2^\omega\to 2^\omega$1 is order-preserving if

$f:2^\omega\to 2^\omega$2

The 2023 paper “Part 1 of Martin’s Conjecture for order-preserving and measure-preserving functions” introduces the class of measure-preserving functions. Degree-theoretically, $f:2^\omega\to 2^\omega$3 is measure-preserving if for every $f:2^\omega\to 2^\omega$4 there is $f:2^\omega\to 2^\omega$5 such that

$f:2^\omega\to 2^\omega$6

Equivalently, $f:2^\omega\to 2^\omega$7 is Martin-above every constant function; and, at the level of degrees, this is equivalent to preserving the Martin measure $f:2^\omega\to 2^\omega$8 under pushforward (Lutz et al., 2023).

The paper proves two structural facts. First, every measure-preserving Turing-invariant function is above the identity on a cone. Second, every order-preserving function is either constant on a cone or measure-preserving. Combining them yields Part I for all order-preserving functions: $f:2^\omega\to 2^\omega$9 For Borel order-preserving functions, the same paper gives a $x\equiv_T y \implies f(x)\equiv_T f(y).$0 proof of Part I; together with the earlier Slaman–Steel theorem that Part II holds for Borel order-preserving functions, this places the order-preserving case very close to complete resolution (Lutz et al., 2023).

Another classical special case is the regressive case. If $x\equiv_T y \implies f(x)\equiv_T f(y).$1 for all $x\equiv_T y \implies f(x)\equiv_T f(y).$2, then under $x\equiv_T y \implies f(x)\equiv_T f(y).$3, Slaman–Steel showed that either $x\equiv_T y \implies f(x)\equiv_T f(y).$4 is constant almost everywhere or $x\equiv_T y \implies f(x)\equiv_T f(y).$5 almost everywhere. Thus a Turing-invariant function that never raises degrees is forced to be either eventually trivial or eventually the identity (Marks et al., 2011).

4. Descriptive set theory, ultrafilters, and reducibility theory

Martin’s conjecture sits at a junction of degree theory and descriptive set theory. Turing invariance is exactly the assertion that a map is a homomorphism from $x\equiv_T y \implies f(x)\equiv_T f(y).$6 to itself, so the conjecture can be regarded as a rigidity statement about self-maps of a countable Borel equivalence relation. This perspective is explicit in work connecting Martin’s conjecture to universality and ergodicity phenomena for countable Borel equivalence relations (Marks et al., 2011).

A major contrast is provided by arithmetic equivalence $x\equiv_T y \implies f(x)\equiv_T f(y).$7, where $x\equiv_T y \implies f(x)\equiv_T f(y).$8 means each is arithmetical in the other. Slaman–Steel proved that arithmetic equivalence is a universal countable Borel equivalence relation. The same survey emphasizes that this universality shows the direct arithmetic analogue of Martin’s conjecture fails: arithmetic-invariant maps can be far more complicated than the Turing-invariant maps predicted by Martin’s framework (Marks et al., 2011).

The ultrafilter formulation introduced in the 2023 measure-preserving paper gives a particularly sharp restatement of Part I. Let $x\equiv_T y \implies f(x)\equiv_T f(y).$9 be the Martin ultrafilter on the Turing degrees. Then Part I is equivalent to the assertion that if $2^\omega$0 is a nonprincipal ultrafilter on $2^\omega$1 with

$2^\omega$2

in the Rudin–Keisler order, then $2^\omega$3. In that form, Part I says that $2^\omega$4 is the unique nonprincipal ultrafilter on the Turing degrees that is Rudin–Keisler-below itself (Lutz et al., 2023).

The local work of 2019 supplies a different descriptive-set-theoretic consequence. For $2^\omega$5, define an equivalence relation $2^\omega$6 on $2^\omega$7 by

$2^\omega$8

Then the map $2^\omega$9 is a Borel reduction from $\equiv_T$0 to computable reducibility $\equiv_T$1 on equivalence relations on $\equiv_T$2. Hence the quasiorder $\equiv_T$3 is at least as complicated as the Turing degrees: $\equiv_T$4 embeds Borelly into $\equiv_T$5, and the quotient by bi-reducibility contains very large chains and antichains (Bard, 2019).

5. Wadge-theoretic refinements

The most systematic refinement of the uniform theory replaces Turing-to-Turing invariance with Turing-to-many-one invariance. In “The uniform Martin’s conjecture for many-one degrees,” Kihara and Montalbán study functions

$\equiv_T$6

that are uniformly invariant from $\equiv_T$7 to many-one equivalence (or, more generally, $\equiv_T$8-many-one equivalence for a better quasi-order $\equiv_T$9). They compare such functions on a cone via the preorder

$\mathcal D$0

For $\mathcal D$1, this is the many-one analogue of the Martin preorder (Kihara et al., 2016).

Their main theorem is that, assuming $\mathcal D$2 for $\mathcal D$3 and $\mathcal D$4 for general bqo $\mathcal D$5, the partial order of $\mathcal D$6-degrees of uniformly $\mathcal D$7-invariant functions under $\mathcal D$8 is isomorphic to the partial order of Wadge degrees of $\mathcal D$9-valued functions on $2^\omega$00. They also show that every uniformly $2^\omega$01-invariant function is $2^\omega$02-equivalent to a uniformly $2^\omega$03-order-preserving one (Kihara et al., 2016).

This result changes the shape of the classification problem. In the Turing-to-Turing setting, the nontrivial hierarchy is jump-like; in the Turing-to-many-one setting, the correct invariant is the full Wadge degree of the associated $2^\omega$04-valued function. The paper explicitly interprets this as a refinement of the uniform Martin conjecture: once the target degrees are many-one rather than Turing, the spectrum of natural uniformly invariant operations is as rich as Wadge theory (Kihara et al., 2016).

Several concrete identifications appear in that framework. The least nontrivial open Wadge degree corresponds to the function sending $2^\omega$05 to a relativized complete c.e. set; the Hausdorff–Kuratowski difference hierarchy on $2^\omega$06 sets corresponds to the Ershov hierarchy of $2^\omega$07 subsets of $2^\omega$08; and, for $2^\omega$09, a complete Wadge degree corresponds to the complement of the hyperjump. These identifications show that the refined theory is not merely formal: it organizes many-one degrees inside Turing degrees by the same structure that organizes definable subsets of Baire space (Kihara et al., 2016).

6. Analogues, counterexamples, and open directions

Beyond the Turing degrees, Martin-style classification becomes more fragile. For the hyperarithmetic degrees, however, a regressive analogue survives. The paper “Martin’s conjecture for regressive functions on the hyperarithmetic degrees” proves that if $2^\omega$10 is hyp-invariant and $2^\omega$11 for all $2^\omega$12, then either $2^\omega$13 is constant on a cone of hyperdegrees or $2^\omega$14 on a cone. A key technical step is a reduction to the case of a continuous function on a pointed perfect tree, suggesting a method that may be reusable in other Martin-style problems (Lutz, 2023).

The enumeration degrees exhibit a much stronger failure of the classical picture. “Martin’s Conjecture in the Enumeration Degrees” shows that, globally, uniformly invariant functions on the enumeration degrees have a much wider range of behaviors than in the Turing degrees. Yet the local theorem is strikingly rigid: on a fixed enumeration degree, a uniformly e-invariant function is either constant there, or it is above the identity or above the skip operator. The same paper also constructs a Borel e-invariant function that is not uniformly e-invariant, giving a degree-structure analogue of a counterexample to Steel’s conjectural uniformization principle (Cordero, 22 Oct 2025).

Several open problems remain central. The full classical conjecture for arbitrary definable Turing-invariant functions is still unresolved. Steel’s conjecture asks whether every Turing-invariant function is equivalent on a cone to a uniformly Turing-invariant one; if true, it would reduce the general problem to the already-understood uniform case (Marks et al., 2011). The 2019 local paper asks whether Part II also has a local origin, formulating a concrete degree-by-degree classification problem for UTI maps on a single degree $2^\omega$15 in terms of a closure class $2^\omega$16 generated by constants, identity, complement, and pointclass jump operators (Bard, 2019).

There are also unresolved determinacy and ultrafilter questions. The measure-preserving approach leaves open whether Part I for Borel measure-preserving functions can be proved using only Borel determinacy, and whether the Rudin–Keisler characterization can be split into weaker independent subproblems. In the ultrafilter language, one would like a sharper description of the structure below the Martin measure $2^\omega$17 and of its relation to other natural ultrafilters such as the Lebesgue and Baire ultrafilters on the Turing degrees (Lutz et al., 2023).

Taken together, these developments support a nuanced view of Martin’s conjecture. On the Turing degrees, substantial fragments of the conjectural hierarchy are now rigidly understood: uniform, regressive, order-preserving, and measure-preserving cases all exhibit the predicted dichotomy, and the uniform theory above the identity is tightly connected to pointclass jumps and Wadge structure. At the same time, extensions to other reducibilities and degree structures show that the conjecture is not merely a generic phenomenon about invariance; it is specific to the fine interaction between Turing reducibility, cones, determinacy, and definability.