Lowering topological entropy over subsets revisited

Let $(X, T)$ be a topological dynamical system. Denote by $h (T, K)$ and $h^B (T, K)$ the covering entropy and dimensional entropy of $K\subseteq X$, respectively. $(X, T)$ is called D-{\it lowerable} (resp. {\it lowerable}) if for each $0\le h\le h (T, X)$ there is a subset (resp. closed subset) $K_h$ with $h^B (T, K_h)= h$ (resp. $h (T, K_h)= h$); is called D-{\it hereditarily lowerable} (resp. {\it hereditarily lowerable}) if each Souslin subset (resp. closed subset) is D-lowerable (resp. lowerable). In this paper it is proved that each topological dynamical system is not only lowerable but also D-lowerable, and each asymptotically $h$-expansive system is D-hereditarily lowerable. A minimal system which is lowerable and not hereditarily lowerable is demonstrated.Comment: All comments are welcome. Transactions of the American Mathematical Society, to appea

Weak expansiveness for actions of sofic groups

In this paper, we shall introduce $h$-expansiveness and asymptotical $h$-expansiveness for actions of sofic groups. By the definitions, each $h$-expansive action of sofic groups is asymptotically $h$-expansive. We show that each expansive action of sofic groups is $h$-expansive, and, for any given asymptotically $h$-expansive action of sofic groups, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy. Observe that asymptotically $h$-expansive property was firstly introduced and studied by Misiurewicz for $\mathbb{Z}$-actions using the language of topological conditional entropy. And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz's ideas when the group is amenable. It turns out that these two definitions are equivalent in this setting.Comment: to appear in Journal of Functional Analysi

Rindler Quantum Gravity

In this note, we explain how asymptotically globally AdS spacetimes can be given an alternate dual description as entangled states of a pair of hyperbolic space CFTs, which are associated with complementary Rindler wedges of the AdS geometry. The reduced density matrix encoding the state of the degrees of freedom in one of these CFTs describes the physics in a single wedge, which we can think of as the region of spacetime accessible to an accelerated observer in AdS. For pure AdS, this density matrix is thermal, and we argue that the microstates in this thermal ensemble correspond to spacetimes that are almost indistinguishable from a Rindler wedge of pure AdS away from the horizon, but with the horizon replaced by some kind of singularity where the geometrical description breaks down. This alternate description of AdS, based on patches associated with particular observers, may give insight into the holographic description of cosmologies where no observer has access to the full spacetime.Comment: 29 pages, 9 figures, PDF LaTe

Tackling information asymmetry in networks: a new entropy-based ranking index

Information is a valuable asset for agents in socio-economic systems, a significant part of the information being entailed into the very network of connections between agents. The different interlinkages patterns that agents establish may, in fact, lead to asymmetries in the knowledge of the network structure; since this entails a different ability of quantifying relevant systemic properties (e.g. the risk of financial contagion in a network of liabilities), agents capable of providing a better estimate of (otherwise) unaccessible network properties, ultimately have a competitive advantage. In this paper, we address for the first time the issue of quantifying the information asymmetry arising from the network topology. To this aim, we define a novel index - InfoRank - intended to measure the quality of the information possessed by each node, computing the Shannon entropy of the ensemble conditioned on the node-specific information. Further, we test the performance of our novel ranking procedure in terms of the reconstruction accuracy of the (unaccessible) network structure and show that it outperforms other popular centrality measures in identifying the "most informative" nodes. Finally, we discuss the socio-economic implications of network information asymmetry.Comment: 12 pages, 8 figure

On distance sets, box-counting and Ahlfors-regular sets

We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent $s>1$. As a corollary, we improve upon a recent result of Orponen, by showing that if $A$ is Ahlfors-regular of dimension $s>1$, then almost all pinned distance sets of $A$ have lower box-counting dimension $1$. We also show that if $A,B\subset\mathbb{R}^2$ have Hausdorff dimension $>1$ and $A$ is Ahlfors-regular, then the set of distances between $A$ and $B$ has modified lower box-counting dimension $1$, which taking $B=A$ improves Orponen's result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.Comment: 22 pages, no figures. v2: added Corollary 1.5 on box dimension of pinned distance sets. v3: numerous fixes and clarifications based on referee report