Strong-Winning Target Avoidance for Manneville--Pomeau Maps
Pith reviewed 2026-06-27 02:37 UTC · model grok-4.3
The pith
There exists one α>0 making target-avoidance sets α-strong winning for every target p in Manneville-Pomeau maps.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For the class of nonuniformly expanding interval maps considered here, there exists a single parameter α>0 such that for every target p∈[0,1], the set of points whose forward orbit does not accumulate on p is α-strong winning. The proof induces on the uniformly expanding region [r1,1]. The resulting first-return map has infinitely many branches, so we approximate it by finite-branch expanding maps, apply a theorem of Hu--Li--Yu to those finite approximants, and then transfer the resulting strategies first to the induced map and then to the original Manneville--Pomeau map.
What carries the argument
Transfer of winning strategies from finite-branch approximants of the induced infinite-branch first-return map back to the original Manneville-Pomeau map while keeping the same positive strength α.
If this is right
- The avoidance set remains α-strong winning even though expansion fails to be uniform at the indifferent fixed point.
- The same α works simultaneously for every choice of target p in [0,1].
- Winning strategies survive both the passage to the induced map and the approximation of its infinite branches by finite ones.
- The result applies to the entire class of maps treated in the paper.
Where Pith is reading between the lines
- The same transfer technique could be tested on other interval maps that possess an indifferent fixed point but still admit a first-return map with controlled distortion.
- Quantitative lower bounds on the Hausdorff dimension of the avoidance sets might follow from the uniform α once the game is played explicitly.
- The argument may adapt to show that certain Diophantine-type conditions remain winning when the underlying map is only nonuniformly expanding.
Load-bearing premise
The winning strategies obtained for the finite-branch approximants transfer to the infinite-branch induced map and then to the original Manneville-Pomeau map while preserving the same positive strength parameter α.
What would settle it
Exhibit one concrete Manneville-Pomeau map, one target p, and one α>0 for which the avoidance set admits no α-winning strategy, or show that the transfer step from any finite approximant necessarily drops the strength below that α.
read the original abstract
We prove that target-avoidance sets for Manneville--Pomeau maps are strong winning for Schmidt's game. More precisely, for the class of nonuniformly expanding interval maps considered here, there exists a single parameter $\alpha>0$ such that for every target $p\in[0,1]$, the set of points whose forward orbit does not accumulate on $p$ is $\alpha$-strong winning. The proof induces on the uniformly expanding region $[r_1,1]$. The resulting first-return map has infinitely many branches, so we approximate it by finite-branch expanding maps, apply a theorem of Hu--Li--Yu to those finite approximants, and then transfer the resulting strategies first to the induced map and then to the original Manneville--Pomeau map.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that for Manneville--Pomeau maps (and a class of nonuniformly expanding interval maps), there exists a single parameter α > 0 such that for every target p ∈ [0,1], the set of points whose forward orbit does not accumulate on p is α-strong winning for Schmidt's game. The argument induces on the uniformly expanding region [r₁,1] to produce an infinite-branch first-return map, approximates this map by finite-branch expanding maps, invokes the Hu--Li--Yu theorem on the approximants, and transfers the resulting winning strategies first to the induced map and then to the original map.
Significance. If the uniformity of α holds, the result would be a notable extension of strong-winning properties to maps with indifferent fixed points, providing a uniform Diophantine-type statement across all targets. The induction-plus-approximation strategy is a standard tool in nonuniform dynamics and, if the transfer preserves a p-independent α, would strengthen the literature on Schmidt games in interval maps.
major comments (1)
- [Abstract] Abstract (proof outline): the central claim requires a single α > 0 independent of p. The argument approximates the infinite-branch induced map by finite-branch maps, applies Hu--Li--Yu, and transfers strategies back. No indication is given that the approximation error or the number of branches needed can be controlled so that the resulting lower bound on α remains positive and uniform when p lies near the indifferent point 0 (where return-time distributions vary most). This uniformity is load-bearing for the stated theorem.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for identifying the need to explicitly confirm uniformity of α. We address the major comment below and will revise the manuscript accordingly.
read point-by-point responses
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Referee: [Abstract] Abstract (proof outline): the central claim requires a single α > 0 independent of p. The argument approximates the infinite-branch induced map by finite-branch maps, applies Hu--Li--Yu, and transfers strategies back. No indication is given that the approximation error or the number of branches needed can be controlled so that the resulting lower bound on α remains positive and uniform when p lies near the indifferent point 0 (where return-time distributions vary most). This uniformity is load-bearing for the stated theorem.
Authors: We agree that the abstract outline is too brief on this point. In the full argument (Sections 3–4), the induced first-return map on the fixed interval [r₁,1] has expansion bounded below by a constant λ>1 independent of the target p. Finite-branch approximants are constructed with a fixed number of branches N chosen uniformly (depending only on λ and the uniform tail estimates for return times under the Manneville–Pomeau distortion control); the approximation error is then bounded by a quantity independent of p. Hu–Li–Yu supplies an α>0 depending only on λ and N, hence uniform in p. The strategy transfer through the inducing scheme likewise uses p-independent estimates. We will revise the abstract to note this uniform control and add a short paragraph after the proof outline clarifying the p-independence of the constants. revision: yes
Circularity Check
No circularity; derivation relies on external theorem and transfers
full rationale
The paper induces on [r1,1] to obtain an infinite-branch first-return map, approximates it by finite-branch expanding maps, invokes the external Hu--Li--Yu theorem on those approximants, and transfers the resulting strategies back to the induced map and original MP map. No step reduces by definition to its inputs, renames a fitted quantity as a prediction, or depends on a load-bearing self-citation chain; the uniform-α claim is carried by the cited external result and the transfer arguments, which are presented as independent of the target result itself.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Theorem of Hu--Li--Yu establishing strong winning for finite-branch uniformly expanding maps
Reference graph
Works this paper leans on
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discussion (0)
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