Subject / 10 records

Measure and ergodic theory

Measure-preserving transformations, joinings, averages, null sets, and measurable groups.

Classification: Second Brain

10 records

• 10.5281/zenodo.238320582026

  Charmenability of Thompson's group F

  Martino Lupini · Thompson groups, C*-algebras, Amenability

  We prove that Thompson's group F and its commutator subgroup F′ are charmenable. The main assertion is that every nonempty compact convex conjugation-invariant family of normalized positive-definite functions on either group contains a trace. Its proof uses commuting groups supported on disjoint intervals and a stationary averaging argument on their joint von Neumann algebra. It is independent of nonamenability of F. The character condition in charmenability follows from the classification of finite factor representations by Dudko and Medynets. We also prove that F is not charfinite and that F′ is charfinite precisely when F is nonamenable. Every nonzero simple quotient of either full group C*-algebra is scalar or is the corresponding reduced group algebra, and every nonzero nuclear quotient admits a multiplicative character.https://github.com/eulogica/charmenability-of-thompsons-group-f

  No files · GitHub named

• 10.5281/zenodo.234517182026

  Simultaneous avoidance of power families of null sequences

  Martino Lupini · Mathematics

  Let (a_n) be a positive sequence tending to zero with liminf a_(n+1)/a_n > 0. For every η ∈ (0,1), we construct a compact set E ⊆ [0,1] of Lebesgue measure greater than 1−η such that {x+s a_n^λ : n ≥ 1} is not contained in E for any x ∈ ℝ, s ≠ 0, and λ > 0. No monotonicity is required, and one set works for all positive powers simultaneously, with infinitely many distinct points of every copy outside E. Taking a_n=e^(−n) gives one compact set of measure arbitrarily close to one containing no infinite geometric progression of any ratio in (0,1), answering negatively a question of Burgin, Goldberg, Keleti, MacMahon and Wang. We also prove a uniform finite hitting statement for ratios in a compact range and scales in [1,2]. The proof adapts random routing to grids shared by all exponents, with a localized count of exponent and scale parameters and a compactness repair for exceptional centres.

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• 10.5281/zenodo.234401542026

  Common null complements for polynomial images of finite sums of perfect sets

  Martino Lupini · Mathematics

  Fix an integer D ≥ 1, and let A₁, …, A₍D₊₂₎ be nonempty compact perfect subsets of the real line. For their Minkowski sum C = A₁ + ⋯ + A₍D₊₂₎, we construct a single null Fσ set B ⊆ ℝ such that P(C) + B = ℝ for every nonconstant real polynomial P of degree at most D. In particular, every sum of three compact perfect sets has one null complement that works under every nonzero dilation. Thus the conclusion of Jung and Lai’s Conjecture 4.8 holds for these sums; the conjecture remains open for arbitrary perfect sets. The conclusion applies to some sums with zero logarithmic Hausdorff and packing dimensions that contain no polynomial image of an infinite summable finite-digit product. The proof isolates a local sum hypothesis that permits the same construction for more general compact sets.

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• 10.5281/zenodo.234257012026

  Pairwise mixing layouts and self-joinings of weakly mixing transformations

  Martino Lupini · Orderability

  Let T be an invertible weakly mixing transformation of a standard probability space (X, μ). We prove that pairwise mixing along any prescribed finite family of integer sequences implies joint mixing: if every two-coordinate marginal of (T^{d₁(b)}x, …, T^{dₖ(b)}x), for x distributed according to μ, converges to μ², then the full distribution converges to μᵏ. No entropy assumption is needed. This answers Ryzhikov's question on mixing sequences. The proof depends essentially on structural results from OpenAI's multiple-mixing preprint. For weakly mixing rank-one transformations, a tower argument shows that every pairwise-independent finite self-joining is product, answering the corresponding questions in Ryzhikov's problem list. Finally, for invertible ergodic transformations of standard nonatomic spaces, two-fold minimal self-joinings imply minimal self-joinings of all finite orders. This follows from OpenAI's theorem and Ryzhikov's equivalence with the restricted multiple-mixing problem.

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• 10.5281/zenodo.233716902026

  Pointwise convergence of bilinear averages for two commuting transformations

  Martino Lupini · Mathematics

  Let T and S be commuting measure-preserving transformations of a probability space. We prove that, for f,g ∈ L³, the bilinear averages (1/N)∑ₙ₌₁ᴺ f(Tⁿx)g(Sⁿx) converge almost everywhere and satisfy an L³ᐟ² bound for their full r-variation for every r > 2. In particular, this answers the bounded-input convergence question recorded as Problem 19 in Frantzikinakis’s 2016 survey, without an invertibility assumption.

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• 10.5281/zenodo.233421262026

  Boundedness exponents of unitary function groups and measured full groups

  Martino Lupini · Descriptive set theory, Groups

  Let $X$ be an infinite compact metrizable space and let $M\ne0$ be a von Neumann algebra with separable predual. For the strong topology on $\mathcal{U}(M)$, we classify boundedness of $C(X,\mathcal{U}(M))$: finite exponents occur exactly when $M$ is properly infinite, with least open-exhaustion exponent five and least finite-neighbourhood exponent three, and ordinary Property $(\mathrm{OB})$ holds exactly when $\dim X=0$ or $M$ has no matrix direct summand. Continuous fixed-involution factors give the finite-exponent result; compact-uniform adaptations of spectral regularization and central splitting give ordinary boundedness. The classification concerns function-group boundedness, while the corresponding target contractibility criterion is due to Jekel. Related four-factor target-unitary and type III full-group bounds are upper bounds.

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• 10.5281/zenodo.232914082026

  Measurable signatures and algebraic structure of L¹ full groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.232914012026

  Local swaps, Polish group topologies, and automorphisms of L¹ full groups

  Martino Lupini · Descriptive set theory, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.232913922026

  Polish topologies and coset obstructions for measurable transformation groups

  Martino Lupini · Descriptive set theory, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228097652026

  Generic monothetic subgroups of the measure automorphism group

  Martino Lupini · Groups

  No abstract on this deposit.

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