Homeomorphic measures on stationary Bratteli diagrams

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation. Equivalently, the set S is formed by ergodic probability measures invariant with respect to aperiodic substitution dynamical systems. The paper is devoted to the classification of measures $\mu$ from S with respect to a homeomorphism. The properties of these measures related to the clopen values set $S(\mu)$ are studied. It is shown that for every measure $\mu$ in S there exists a subgroup G of $\mathbb R$ such that $S(\mu)$ is the intersection of G with [0,1], i.e. $S(\mu)$ is group-like. A criterion of goodness is proved for such measures. This result is used to classify the measures from S up to a homeomorphism. It is proved that for every good measure $\mu$ in S there exist countably many measures $\{\mu_i\}_{i\in \mathbb N}$ from S such that $\mu$ and $\mu_i$ are homeomorphic measures but the tail equivalence relations on corresponding Bratteli diagrams are not orbit equivalent.Comment: 36 pages, references added, typos fixe

Subdiagrams of Bratteli diagrams supporting finite invariant measures

We study finite measures on Bratteli diagrams invariant with respect to the tail equivalence relation. Amongst the proved results on finiteness of measure extension, we characterize the vertices of a Bratteli diagram that support an ergodic finite invariant measure.Comment: 9 page

Invariant Measures on Stationary Bratteli Diagrams

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we explicitly describe all ergodic probability measures invariant with respect to the tail equivalence relation (or the Vershik map). These measures are completely described by the incidence matrix of the diagram. Since such diagrams correspond to substitution dynamical systems, this description gives an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.Comment: 40 pages. Exposition is reworke

Spatially-resolved probing of a non-equilibrium superconductor

Spatially resolved relaxation of non-equilibrium quasiparticles in a superconductor at ultra-low temperatures was experimentally studied. It was found that the quasiparticle injection through a tunnel junction results in modification of the shape of I-V characteristic of a remote `detector' junction. The effect depends on temperature, injection current and proximity to the injector. The phenomena can be understood in terms of creation of quasiparticle charge and energy disequilibrium characterized by two different length scales $\Lambda_{Q^{\ast}}$ $\sim5$ $\mu$m and $\Lambda_{T^{\ast}}\sim$ $40$ $\mu$m. The findings are in good agreement with existing phenomenological models, while more elaborated microscopic theory is mandatory for detailed quantitative comparison with experiment. The results are of fundamental importance for understanding electron transport phenomena in various nanoelectronic circuits.Comment: 7 pages, 5 figure

Perfect orderings on finite rank Bratteli diagrams

Given a Bratteli diagram B, we study the set OB of all possible orderings on B and its subset PB consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram B. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram B of rank k, we endow the set OB with product measure µ and prove that there is some 1 ≤ j ≤ k such that µalmost all orderings on B have j maximal and j minimal paths. If j is strictly greater than the number of minimal components that B has, then µ-almost all orderings are imperfect

Current-voltage characteristics of asymmetric double-barrier Josephson junctions

We develop a theory for the current-voltage characteristics of diffusive superconductor-normal metal-superconductor Josephson junctions with resistive interfaces and the distance between the electrodes smaller than the superconducting coherence length. The theory allows for a quantitative analytical and numerical analysis in the whole range of the interface transparencies and asymmetry. We focus on the regime of large interface resistance compared to the resistance of the normal region, when the electron-hole dephasing in the normal region is significant and the finite length of the junction plays a role. In the limit of strong asymmetry we find pronounced current structures at the combination subharmonics of $\Delta+\Delta_g$, where $\Delta_g$ is the proximity minigap in the normal region, in addition to the subharmonics of the energy gap $2\Delta$ in the electrodes. In the limit of rather transparent interfaces, our theory recovers a known formula for the current in a short mesoscopic connector - a convolution of the current through a single-channel point contact with the transparency distribution for an asymmetric double-barrier potential.Comment: 10 pages, 3 figure