The strip entropy approximation of Markov shifts on trees

The strip entropy is studied in this article. We prove that the strip entropy approximation is valid for every ray of a golden-mean tree. This result extends the previous result of [Petersen-Salama, Discrete \& Continuous Dynamical Systems, 2020] on the conventional 2-tree. Lastly, we prove that the strip entropy approximation is valid for eventually periodic rays of a class of Markov-Cayley trees

Thermally-Switchable Metalenses Based on Quasi-Bound States in the Continuum

Dynamic wavefront shaping with optical metasurfaces has presented a major challenge and inspired a large number of highly elaborate solutions. Here, we experimentally demonstrate thermo-optically reconfigurable, nonlocal metasurfaces using simple device architectures and conventional CMOS-compatible dielectric materials. These metasurfaces support quasi-bound states in the continuum (q-BICs) derived from symmetry breaking and encoded with a spatially varying geometric phase, such that they shape optical wavefront exclusively on spectrally narrowband resonances. Due to the enhanced light-matter interaction enabled by the resonant q-BICs, a slight variation of the refractive index introduced by heating and cooling the entire device leads to a substantial shift of the resonant wavelength and a subsequent change to the optical wavefront associated with the resonance. We experimentally demonstrate a metalens modulator, the focusing capability of which can be thermally turned on and off, and reconfigurable metalenses, which can be thermo-optically switched to produce two distinct focal patterns. Our devices offer a pathway to realize reconfigurable, multifunctional meta-optics using established manufacturing processes and widely available dielectric materials that are conventionally not considered "active" materials due to their small thermo-optic or electro-optic coefficients