Exponential mixing property for Hénon–Sibony maps of

Let f be a Hénon–Sibony map, also known as a regular polynomial automorphism of $\mathbb {C}^k$ , and let $\mu $ be the equilibrium measure of f. In this paper we prove that $\mu $ is exponentially mixing for plurisubharmonic observables.

Simple conditions for metastability of continuous Markov chains

A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

The multi-transitivity of free semigroup actions

In this paper, we introduce the conceptions of multi-transitivity, [Formula: see text]-transitivity and [Formula: see text]-mixing property for free semigroup actions and give some equivalent conditions for a free semigroup action to be multi-transitive, multi-transitive with respect to vectors and strongly multi-transitive, respectively. For instance, we prove that a free semigroup action is multi-transitive or multi-transitive with respect to a vector if and only if its corresponding skew product system is multi-transitive or multi-transitive with respect to the same vector.

Chaotic Binary Sequence Generator based on Logistic Map

Pseudorandom binary sequences find various applications in different areas such as security, communication, steganography and cryptography. The properties like sensitivity to initial condition, ergodicity, mixing property and dynamic behavior are used in the designing of random number generators known as chaotic systems. In this study, an efficient chaotic binary sequence generator using logistic map is proposed, implemented and analyzed. The proposed binary sequence generator generates 50 chaotic sequences by varying initial condition with fixed system parameter. Subsequently, the generated sequences are transformed to binary sequences using thresholding method. The output of binary sequences is statistically tested with FIPS 140-2 test suite in order to identify the specific properties expected for truly random binary sequences. The experimental results prove that the generated binary sequences possess identical characteristics of true random numbers and can pass all tests of FIPS 140-2 test suite.