Tree pressure for hyperbolic and non-exceptional upper semi-continuous potentials

In this note, we investigate the tree pressure for multi-modal interval maps with a certain class of hyperbolic and non-exceptional upper semi-continuous functions. In particular, we obtain a generalized version of Corollary 2.2 in the paper \cite{LRL14} by Li and Rivera-Letelier. This property will be used to prove the existence of a conformal measure for the geometric potential in the negative spectrum.Comment: This is a technical note and not intended for publication. arXiv admin note: text overlap with arXiv:1210.6952 by other author

Thermodynamic formalism of interval maps for upper semi-continuous potentials: Makarov-Smirnov's formalism

In this paper, we study the thermodynamic formalism of interval maps $f$ with sufficient regularity, for a sub class $\mathcal{U}$ composed of upper semi-continuous potentials which includes both H\"{o}lder and geometric potentials. We show that for a given $u\in \mathcal{U}$ and negative values of $t$, the pressure function $P(f,-tu)$ can be calculated in terms of the corresponding hidden pressure function $\widetilde{P}(f,-tu)$. Determination of the values $t\in(-\infty,0)$ at which $P(f,-tu)\neq \widetilde{P}(f,-tu)$ is also characterized explicitly. When restricting to the H\"{o}lder continuous potentials, our result recovers Theorem B in [Li \& Rivera-Letelier 2013] for maps with non-flat critical points. While restricting to the geometric potentials, we develop a real version of Makarnov-Smirnov's formalism, in parallel to the complex version shown in [Makarnov \& Smirnov 2000, Theo A,B]. Moreover, our results also provide a new and simpler proof (using [Ruelle 1992, Coro6.3]) of the original Makarnov-Smirnov's formalism in the complex setting, under an additional assumption about non-exceptionality, i.e., [Makarnov \& Smirnov 2000, Theo 3.1].Comment: 41 pages, 1 figure. arXiv admin note: text overlap with arXiv:1210.0521 by other author

Invariant Measures with Bounded Variation Densities for Piecewise Area Preserving Maps

We investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on $\mathbb{R}^d$. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results "partially" answer one of the fundamental questions posed in \cite{Goetz03} - to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e., they have unbounded variation.Comment: 19 page

Private Information Retrieval from MDS Coded Databases with Colluding Servers under Several Variant Models

Private information retrieval (PIR) gets renewed attentions due to its information-theoretic reformulation and its application in distributed storage system (DSS). The general PIR model considers a coded database containing $N$ servers storing $M$ files. Each file is stored independently via the same arbitrary $(N,K)$-MDS code. A user wants to retrieve a specific file from the database privately against an arbitrary set of $T$ colluding servers. A key problem is to analyze the PIR capacity, defined as the maximal number of bits privately retrieved per one downloaded bit. Several extensions for the general model appear by bringing in various additional constraints. In this paper, we propose a general PIR scheme for several variant PIR models including: PIR with robust servers, PIR with Byzantine servers, the multi-file PIR model and PIR with arbitrary collusion patterns.Comment: The current draft is extended by considering several PIR models. The original version named "Multi-file Private Information Retrieval from MDS Coded Databases with Colluding Servers" is abridged into a section within the current draft. arXiv admin note: text overlap with arXiv:1704.0678

Ergodic optimization of prevalent super-continuous functions

Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this paper we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph.Comment: 23 pages, 4 figures. Final version, to appear in International Mathematics Research Notices (IMRN

Dimension results for inhomogeneous Moran set constructions

We compute the Hausdorff, upper box and packing dimensions for certain inhomogeneous Moran set constructions. These constructions are beyond the classical theory of iterated function systems, as different nonlinear contraction transformations are applied at each step. Moreover, we also allow the contractions to be weakly conformal and consider situations where the contraction rates have an infimum of zero. In addition, the basic sets of the construction are allowed to have a complicated topology such as having fractal boundaries. Using techniques from thermodynamic formalism we calculate the fractal dimension of the limit set of the construction. As a main application we consider dimension results for stochastic inhomogeneous Moran set constructions, where chaotic dynamical systems are used to control the contraction factors at each step of the construction.Comment: 30pages and 1 figur

Snake-in-the-Box Codes for Rank Modulation under Kendall's τ\tau-Metric

For a Gray code in the scheme of rank modulation for flash memories, the codewords are permutations and two consecutive codewords are obtained using a push-to-the-top operation. We consider snake-in-the-box codes under Kendall's $\tau$-metric, which is a Gray code capable of detecting one Kendall's $\tau$-error. We answer two open problems posed by Horovitz and Etzion. Firstly, we prove the validity of a construction given by them, resulting in a snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+1$. Secondly, we come up with a different construction aiming at a longer snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+3$. The construction is applied successfully to $S_7$.Comment: arXiv admin note: text overlap with arXiv:1311.4703 by other author

A general private information retrieval scheme for MDS coded databases with colluding servers

The problem of private information retrieval gets renewed attentions in recent years due to its information-theoretic reformulation and applications in distributed storage systems. PIR capacity is the maximal number of bits privately retrieved per one bit of downloaded bit. The capacity has been fully solved for some degenerating cases. For a general case where the database is both coded and colluded, the exact capacity remains unknown. We build a general private information retrieval scheme for MDS coded databases with colluding servers. Our scheme achieves the rate $(1+R+R^2+\cdots+R^{M-1})$, where $R=1-\frac{{{N-T}\choose K}}{{N\choose K}}$. Compared to existing PIR schemes, our scheme performs better for a certain range of parameters and is suitable for any underlying MDS code used in the distributed storage system.Comment: Submitted to IEEE Transactions on Information Theor

A rigorous computer aided estimation for Gelfond exponent of weighted Thue-Morse sequences

In this paper, we will provide a mathematically rigorous computer aided estimation for the exact values and robustness for Gelfond exponent of weighted Thue-Morse sequences. This result improves previous discussions on Gelfond exponent by Gelfond, Devenport, Mauduit, Rivat, S\'{a}rk\"{o}zy and Fan et. al.Comment: 22 pages, 7 Figures, 4 Table

Invertible binary matrix with maximum number of 22-by-22 invertible submatrices

The problem is related to all-or-nothing transforms (AONT) suggested by Rivest as a preprocessing for encrypting data with a block cipher. Since then there have been various applications of AONTs in cryptography and security. D'Arco, Esfahani and Stinson posed the problem on the constructions of binary matrices for which the desired properties of an AONT hold with the maximum probability. That is, for given integers $t\le s$, what is the maximum number of $t$-by-$t$ invertible submatrices in a binary matrix of order $s$? For the case $t=2$, let $R_2(s)$ denote the maximal proportion of 2-by-2 invertible submatrices. D'Arco, Esfahani and Stinson conjectured that the limit is between 0.492 and 0.625. We completely solve the case $t=2$ by showing that $\lim_{s\rightarrow\infty}R_2(s)=0.5$