Dense Gδ Research Articles

Polynomial orbits in totally minimal systems

Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye [14], we prove that the maximal ∞-step pro-nilfactor X∞ of a minimal system (X,T) is the topological characteristic factor along polynomials in a certain sense. Namely, we show that by an almost one to one modification of π:X→X∞, the induced open extension π⁎:X⁎→X∞⁎ has the following property: for any d∈N, any open subsets V0,V1,…,Vd of X⁎ with ⋂i=0dπ⁎(Vi)≠∅ and any distinct non-constant integer polynomials pi with pi(0)=0 for i=1,…,d, there exists some n∈Z such that V0∩T−p1(n)V1∩…∩T−pd(n)Vd≠∅.As an application, the following result is obtained: for a totally minimal system (X,T) and integer polynomials p1,…,pd, if every non-trivial integer combination of p1,…,pd is not constant, then there is a dense Gδ subset Ω of X such that the set{(Tp1(n)x,…,Tpd(n)x):n∈Z} is dense in Xd for every x∈Ω.

On comeager sets of metrics whose ranges are disconnected

For a metrizable space X, we denote by Met(X) the space of all metric that generate the same topology of X. The space Met(X) is equipped with the supremum distance. In this paper, for every strongly zero-dimensional metrizable space X, we prove that the set of all metrics whose ranges are closed totally disconnected subsets of the line is a dense Gδ subspace in Met(X). As its application, we show that some sets of universal metrics are meager in spaces of metrics.

Extension of mappings from the product of pseudocompact spaces

Let X and Y be pseudocompact spaces and let a function Φ:X×Y→R be separately continuous. The following conditions are equivalent: (1) there is a dense Gδ subset of D⊂Y such that Φ is continuous at every point of X×D (Namioka property); (2) Φ is quasicontinuous; (3) Φ extends to a separately continuous function on βX×βY. This theorem makes it possible to combine studies of the Namioka property and generalizations of the Eberlein-Grothendieck theorem on the precompactness of subsets of function spaces. We also obtain a characterization of separately continuous functions on the product of several pseudocompact spaces extending to separately continuous functions on products of Stone–Čech extensions of spaces. These results are used to study groups and Mal'tsev spaces with separately continuous operations.

On continuity of set-valued mappings

We will show that under certain circumstances upper and lower quasi-continuity implies continuity. We also extend Fort's theorem for upper quasi-continuous mappings, by showing that every upper quasi-continuous mapping from a Baire space into compact subsets of a regular locally metrizable space is continuous at points of a dense Gδ subset of X.

Joint continuity of separately continuous mappings

In this paper, we will introduce a topological game to show that every separately continuous function f:X×Y→Z, where X is a β-unfavorable space for our game and Z is a regular space, is strongly quasi-continuous at each point of X×{y} provided that y is a q-point. We also show that the set of points of subcontinuity of quasi-continuous mappings from a Baire space X into a regular game determined space Y contains a dense Gδ subset of X. Our results generalize some theorems in Bouziad (1996) [6] and Kenderov et al. (2001) [18].

Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game);The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X).In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset;the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X).If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).

Structural instability of semi-Siegel Hénon maps

We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not J-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense Gδ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.

Dynamics of measurable functions on the interval

Let A⊂I=[0,1] such that λA=0 and A is a dense Gδ subset of [0,1], and take M to be the set of measurable self-maps of I. There exists a residual set R⊂M such that for each f in R, the following hold:(1)The range of f is contained in A, and f is a one-to-one function.(2)For any x∈[0,1], the ω-limit set ω(x,f) is nowhere dense.(3)The Hausdorff dimension of the ω-limit points Λ(f)‾=∪x∈Iω(x,f) is zero.(4)The function f is nowhere continuous.Any closed set E contained in [0,1] is an ω-limit set for some measurable function f:I→I. Moreover, there exists a measurable function f:[0,1]→[0,1] such that for any ε>0, x∈[0,1] and closed set E⊂[0,1], there is a function g:I→I such that ∥g−f∥<ε, and E=ω(x,g).

Phase Transition of Logarithmic Capacity for the Uniform Gδ-Sets

We consider a family of dense Gδ subsets of [0, 1], defined as intersections of unions of small uniformly distributed intervals, and study their logarithmic capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a Gδ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. Our re-distribution construction can be considered as a generalization of a method applied by Ursell in his construction of a counter-example to a conjecture by Nevanlinna. Also, we propose a simple Cauchy-Schwartz inequality-based proof of related theorems by Lindeberg and by Erdos and Gillis.

Generic zero-Hausdorff and one-packing spectral measures

For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneous zero upper-Hausdorff and one lower-packing dimension contains a dense Gδ subset. Applications include sets of limit-periodic operators.

Parametric Furstenberg Theorem on random products of [formula omitted] matrices

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

Small values of Weyl sums

We prove that the set of (x1,…,xd)∈[0,1)d, such thatlim_N→∞|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|=0, contains a dense Gδ set in [0,1)d and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums∑n=1Nexp⁡(2πixnd),x∈[0,1).

A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli

Let M be a smooth compact connected manifold of dimension d ≥ 2, possibly with boundary, that admits a smooth effective $${\mathbb{T}^2}$$ -action $${\cal S} = {\left\{{{S_{\alpha ,\beta}}} \right\}_{\left({\alpha ,\beta} \right) \in {\mathbb{T}^2}}}$$ preserving a smooth volume v, and let $${\cal B}$$ be the C∞ closure of $$\left\{{h\, \circ {S_{\alpha ,\beta}} \circ {h^{- 1}}:h \in {\rm{Dif}}{{\rm{f}}^\infty}\left({M,v} \right),\left({\alpha ,\beta} \right) \in {\mathbb{T}^2}} \right\}.$$ We construct a weakly mixing C∞ diffeomorphism $$T \in {\cal B}$$ with topological entropy 0 such that T × T is loosely Bernoulli. Moreover, we show that the set of such $$T \in {\cal B}$$ contains a dense Gδ subset of $${\cal B}$$ . The proofs are based on a two-dimensional version of the approximation-by-conjugation method.

Preimage pressure, stable pressure and equilibrium states

In this paper, the pressure of a potential via the dynamics of preimage structure are investigated for noninvertible systems and the properties for the related equilibrium states are considered. For a continuous map on a compact metric space, a notion of preimage pressure via the preimages of single points is introduced, and then some fundamental properties are considered and a variational principle is established for the system with uniform separation of preimages. For a C1-partially hyperbolic endomorphism on a closed Riemannian manifold, a notion of stable pressure via the preimages of local stable manifolds is introduced and a corresponding variational principle is established. Moreover, equilibrium states and tangent functionals for pointwise preimage pressures and stable pressures are also considered. By describing the set of equilibrium states and tangent functionals and a sense of continuous dependence of equilibrium states with respect to the potential, we prove that the continuous potentials with unique equilibrium state form a dense Gδ set. In addition, for any finite collection of ergodic measures, we can find a continuous potential such that its set of equilibrium states contains the given set.

All projections of a typical Cantor set are Cantor sets

In 1994, John Cobb asked: given N>m>k>0, does there exist a Cantor set in RN such that each of its projections into m-planes is exactly k-dimensional? Such sets were described for (N,m,k)=(2,1,1) by L. Antoine (1924) and for (N,m,m) by K. Borsuk (1947). Examples were constructed for the cases (3,2,1) by J. Cobb (1994), for (N,m,m−1) and in a different way for (N,N−1,N−2) by O. Frolkina (2010, 2019), for (N,N−1,k) by S. Barov, J.J. Dijkstra and M. van der Meer (2012). We show that such sets are exceptional in the following sense. Let C(RN) be the set of all Cantor subsets of RN endowed with the Hausdorff metric. It is known that C(RN) is a Baire space. We prove that there is a dense Gδ subset P⊂C(RN) such that for each X∈P and each non-zero linear subspace L⊂RN, the orthogonal projection of X into L is a Cantor set. This gives a partial answer to another question of J. Cobb stated in the same paper (1994).

Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

In this paper, we prove that if C⁎⁎ is a ε-separable bounded subset of X⁎⁎, then every convex function g≤σC is Ga^teaux differentiable at a dense Gδ subset G of X⁎ if and only if every subset of ∂σC(0)∩X is weakly dentable. Moreover, we also prove that if C is a closed convex set, then dσC(x⁎)=x if and only if x is a weakly exposed point of C exposed by x⁎. Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C⁎ of X⁎, there exists a dense subset G of X⁎⁎ such that σC⁎ is Ga^teaux differentiable on G and dσC⁎(G)⊂C⁎. We also prove that X is an Asplund space if and only if, for every w⁎-lower semicontinuous convex function f, there exists a dense subset G of X⁎⁎ such that f is Ga^teaux differentiable on G and df(G)⊂X⁎.

On irreducible operators in factor von Neumann algebras

Let M be a factor von Neumann algebra with separable predual and let T∈M. We call T an irreducible operator (relative to M) if W⁎(T) is an irreducible subfactor of M, i.e., W⁎(T)′∩M=CI. In this note, we show that the set of irreducible operators in M is a dense Gδ subset of M in the operator norm. This is a natural generalization of a theorem of Halmos.

Points of Upper and Lower Semicontinuity for Multivalued Functions

We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely, among other results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping F : X × Y → K(Z) is jointly upper semicontinuous on sets of the form D × {y0}, where D is a dense G𝛿 -subset of X and y0 ∈ Y. A similar result was obtained for the joint lower semicontinuity of upper horizontally quasicontinuous mappings. These results improve some known results on the joint continuity of single-valued functions.

On minimal homeomorphisms on Peano continua

Following the question of Artigue we construct a minimal homeomorphism g: X→X on a Peano continuum X with the following property: there exist a positive number ε and a dense Gδ subset E of X such that every non-trivial subcontinuum of X intersecting E expands under iterations of g to a continuum of diameter greater than ε.

Insecurity is generic in a conformal class of Riemannian metrics

A pair of points x, y in a Riemannian manifold (M,g) is said to be secure if there exists a finite set of points intercepting every geodesic segment joining x to y. Given any conformal equivalence class C of Riemannian metrics on a closed manifold M of dimension at least two and given any pair of points x, y in M, there exists a dense Gδ set C′⊂C such that x and y are not secure for every metric g in C′.