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A group G is said to be maximally almost periodic if there is an injective homomorphism from G into a compact Hausdorff group. … My question is whether every discrete countable FC-group G is maximally almost periodic. …

If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is injective …

.$$ Now consider an almost periodic function $f$. … It is easy to see that hypothesis 1 is true for the periodic case. But I even don't know what to do with special almost periodic case. For the general case, i.e. …

The buzzword to look for is "maximally almost periodic". This is part of the theory of Bohr Compactifications. Start with a (locally compact) group $G$. … By definition, this means that $G$ is "maximally almost periodic" (MAP). …

A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) norm topology. … Then $f$ is uniformly continuous with respect to the left and right uniform structures on $G$ and all (weakly) almost periodic functions forms an abelian $C^*$-algebra with the norm of uniform convergence …

According to Proposition 3.3 from Dikranjan and Toller [Topology and its Applications 159 (2012) 2951-2972], the Heisenberg group H_K over an infinite field K of characteritic 0 is not maximally almost … periodic. …

Does there exist a finite set of radii such that some aperiodic packing of the plane by disks of those radii is believed to achieve the maximal packing density (not achieved by any periodic packing)? … [Note to administrators: I wanted to add “aperiodic” or “quasicrystal” or something like that as a tag; I settled for “almost-periodic-function”, but please re-tag as appropriate.] …

As a complement to Reid's answer: a finitely generated group is maximally almost periodic if and only if it is residually finite. …

Such groups are called "maximally almost periodic". Take a look at this paper for a more in-depth treatment. …

Can I deform the pair $(\alpha, \omega)$ so that the only minimal invariant sets of the resulting vector field are non-degenerate periodic orbits? … I am now allowing a larger family of deformations (since the 2-form is allowed to vary among maximally non-degenerate 2-forms, dropping the condition of being closed.) …

Note: I have restricted my answer to abelian $G$; $\iota$ can be defined for any topological group, but your question only makes sense when $\iota$ is injective, i.e. for “maximally almost-periodic” $G …

The maximal ideal space of the set $AP(\mathbb R)$ of almost periodic functions. i.e the set of multiplicative linear functionals under the pointwise topology. …

A function $u\in C_{t,loc}^{0}L_{x}^{2}(I\times\mathbb{R}^{N})$ is almost periodic modulo $G$ if the quotiented orbit $\{Gu(t) : t\in I\}$ is a precompact subset of $G\backslash L_{x}^{2}(\mathbb{R}^{N … Equivalently, $u$ is almost periodic modulo $G$ if there exists a compact subset $K\subset L_{x}^{2}(\mathbb{R}^{N})$ such that $u(t)\in GK$, $t\in I$. …

The maximal open set $U$ with this property is called the basin of $f$. It is well known that every transitive Anosov diffeomorphism carries a unique SRB measure. … My solution : Let $p \in M$ be a periodic point associated to $\delta, S$ and $\{x, f(x),..., f^{n-1}(x)\}$ in the periodic exponential specification property. …

unfortunate, as $u$ is not a compactification in the sense usually used in topology, unless $G$ was compact to begin with (in fact, $u$ is rarely injective - groups for which $u$ is injective are called "maximally … almost periodic"). …