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We prove that the Sierpinski curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowens specification property.
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Deterministic fractal antennas are employed to realize multimodal plasmonic devices. Such structures show strongly enhanced localized electromagnetic fields typically in the infrared range with a hierarchical spatial distribution. Realization of engineered fractal antennas operating in the optical regime would enable nanoplasmonic platforms for applications, such as energy harvesting, light sensing, and bio/chemical detection. Here, we introduce a novel plasmonic multiband metamaterial based on the Sierpinski carpet (SC) space-filling fractal, having a tunable and polarization-independent optical response, which exhibits multiple resonances from the visible to mid-infrared range. We investigate gold SCs fabricated by electron-beam lithography on CaF$_{2}$ and Si/SiO$_{2}$ substrates. Furthermore, we demonstrate that such resonances originate from diffraction-mediated localized surface plasmons, which can be tailored in deterministic fashion by tuning the shape, size, and position of the fractal elements. Moreover, our findings illustrate that SCs with high order of complexity present a strong and hierarchically distributed electromagnetic near-field of the plasmonic modes. Therefore, engineered plasmonic SCs provide an efficient strategy for the realization of compact active devices with a strong and broadband spectral response in the visible/mid-infrared range. We take advantage of such a technology by carrying out surface enhanced Raman spectroscopy (SERS) on Brilliant Cresyl Blue molecules deposited onto plasmonic SCs. We achieve a broadband SERS enhancement factor up to $10^{4}$, thereby providing a proof-of-concept application for chemical diagnostics.
Recent advances in nanofabrication methods have made it possible to create complex two-dimensional artificial structures, such as fractals, where electrons can be confined. The optoelectronic and plasmonic properties of these exotic quantum electron systems are largely unexplored. In this article, we calculate the optical conductivity of a two-dimensional electron gas in a Sierpinski carpet (SC). The SC is a paradigmatic fractal that can be fabricated in a planar solid-state matrix by means of an iterative procedure. We show that the optical conductivity as a function of frequency (i.e. the optical spectrum) converges, at finite temperature, as a function of the fractal iteration. The calculated optical spectrum features sharp peaks at frequencies determined by the smallest geometric details at a given fractal iteration. Each peak is due to excitations within sets of electronic state-pairs, whose wave functions are characterized by quantum confinement in the SC at specific length scales, related to the frequency of the peak.
We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}le b_{ij}le c_i<1. By doing so we show that the variational principle for the dimension holds for this class.
In this paper we give an elementary treatment of the dynamics of skew tent maps. We divide the two-parameter space into six regions. Two of these regions are further subdivided into infinitely many regions. All of the regions are given explicitly. We find the attractor in each subregion, determine whether the attractor is a periodic orbit or is chaotic, and also determine the asymptotic fate of every point. We find that when the attractor is chaotic, it is either a single interval or the disjoint union of a finite number of intervals; when it is a periodic orbit, all periods are possible. Sometimes, besides the attractor, there exists an invariant chaotic Cantor set.
We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket $SG_{2,b}(n)$ at stage $n$ with $b$ equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants for $SG_{2,b}$ and $d$-dimensional Sierpinski gasket $SG_d$.
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