The Chebyshev map is a typical chaotic map. We consider generalized Chebyshev maps Tk of C2. The support of the maximal entropy measure of Tk is connected. We perturb Tk in a certain direction. Then we can show the support of the maximal entropy measure of this map is a Cantor set.

Let K be a field of characteristic p > 0. It is proved that each automorphism σ ∈ AutK(D(Pn)) of the ring D(Pn) of differential operators on a polynomial algebra Pn = K[x1, . . . , xn] is uniquely determined by the elements σ(x1), . . . , σ(xn), and the set Frob(D(Pn)) of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of D(Pn), like Pn, is equal to A...

We examine the measurable ergodic theory of analytic maps F of complex projective space. We focus on two different classes of maps, Ueda maps of Pn, and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weakor strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss t...

The article discusses two mutually-incompatible hypotheses about the stochastic mechanism of the generation of texts in natural language, which could be related to entropy. The first hypothesis, the finite energy hypothesis, assumes that texts are generated by a process with exponentially-decaying probabilities. This hypothesis implies a logarithmic upper bound for maximal repetition, as a func...

This paper compares a number of centrality measures and several (dis-)similarity matrices with which they can be defined. These matrices, which are used among others in community detection methods, represent quantities connected to enumeration of paths on a graph and to random walks. Relationships between some of these matrices are derived in the paper. These relationships are inherited by the ...

We address the problem of estimating the running quantile of a data stream when the memory for storing observations is limited. We (i) highlight the limitations of approaches previously described in the literature which make them unsuitable for non-stationary streams, (ii) describe a novel principle for the utilization of the available storage space, and (iii) introduce two novel algorithms whi...

We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk loca...

We quantify the measurement-induced nonlocality (Luo and Fu, Phys. Rev. Lett. 106, 1020401, 2011) from the perspective of the relative entropy. This quantification leads to an operational interpretation for the measurement-induced nonlocality, namely, it is the maximal entropy increase after the locally invariant measurements. The relative entropy of nonlocality is upper bounded by the entropy ...

We consider the problem of reconstructing global quantum states from local data. Because the reconstruction problem has many solutions in general, we consider the reconstructed state of maximum global entropy consistent with the local data. We show that unique ground states of local Hamiltonians are exactly reconstructed as the maximal entropy state. More generally, we show that if the state in...