On the multifractal spectrum of weighted Birkhoff averages

<p style='text-indent:20px;'>In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.</p>

A new proof of the dimension gap for the Gauss map

In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a ‘dimension gap’ meaning that for some c > 0, sup p dim μ p < 1– c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

Intrinsic density, asymptotic computability, and stochasticity

There are many computational problems which are generally “easy” to solve but have certain rare examples which are much more difficult to solve. One approach to studying these problems is to ignore the difficult edge cases. Asymptotic computability is one of the formal tools that uses this approach to study these problems. Asymptotically computable sets can be thought of as almost computable sets, however every set is computationally equivalent to an almost computable set. Intrinsic density was introduced as a way to get around this unsettling fact, and which will be our main focus.Of particular interest for the first half of this dissertation are the intrinsically small sets, the sets of intrinsic density $0$ . While the bulk of the existing work concerning intrinsic density was focused on these sets, there were still many questions left unanswered. The first half of this dissertation answers some of these questions. We proved some useful closure properties for the intrinsically small sets and applied them to prove separations for the intrinsic variants of asymptotic computability. We also completely separated hyperimmunity and intrinsic smallness in the Turing degrees and resolved some open questions regarding the relativization of intrinsic density.For the second half of this dissertation, we turned our attention to the study of intermediate intrinsic density. We developed a calculus using noncomputable coding operations to construct examples of sets with intermediate intrinsic density. For almost all $r\in (0,1)$ , this construction yielded the first known example of a set with intrinsic density r which cannot compute a set random with respect to the r-Bernoulli measure. Motivated by the fact that intrinsic density coincides with the notion of injection stochasticity, we applied these techniques to study the structure of the more well-known notion of MWC-stochasticity.Abstract prepared by Justin Miller.E-mail: [email protected]: https://curate.nd.edu/show/6t053f4938w

The Entropy and Reversibility of Cellular Automata on Cayley Tree

In this paper, we study linear cellular automata (CAs) on Cayley tree of order [Formula: see text] over the field [Formula: see text] (the set of prime numbers modulo [Formula: see text]). After revealing the rule matrix corresponding to cellular automata on Cayley tree with the null boundary condition, we analyze the reversibility problem of these cellular automata for some given values of [Formula: see text] and the levels [Formula: see text] of Cayley tree. The necessary and sufficient conditions for determining whether a CA is reversible or not are demonstrated. Furthermore, we compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree, the measure entropy with respect to uniform Bernoulli measure is infinity.

A von Neumann algebraic approach to self-similar group actions

We study some relations between self-similar group actions and operator algebras. We see that [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the Bernoulli measure and [Formula: see text] the set of [Formula: see text]-generic points. In the case [Formula: see text], we get a unique KMS state for the canonical gauge action on the Cuntz–Pimsner algebra constructed from a self-similar group action by Nekrashevych. Moreover, if [Formula: see text], there exists a unique tracial state on the gauge invariant subalgebra of the Cuntz–Pimsner algebra. We also consider the GNS representation of the unique KMS state and compute the type of the associated von Neumann algebra.

Effective aspects of Bernoulli randomness

In this paper, we study Bernoulli random sequences, i.e. sequences that are Martin-Löf random with respect to a Bernoulli measure $\mu _p$ for some $p\in [0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in particular on the case in which the underlying Bernoulli parameter $p$ is proper (i.e. Martin-Löf random with respect to some computable measure). We show for every Bernoulli parameter $p$, if there is a sequence that is both proper and Martin-Löf random with respect to $\mu _p$, then $p$ itself must be proper, and explore further consequences of this result. We also study the Turing degrees of Bernoulli random sequences, showing, for instance, that the Turing degrees containing a Bernoulli random sequence do not coincide with the Turing degrees containing a Martin-Löf random sequence. Lastly, we consider several possible approaches to characterizing blind Bernoulli randomness, where the corresponding Martin-Löf tests do not have access to the Bernoulli parameter $p$, and show that these fail to characterize blind Bernoulli randomness.

Existence of an Unbiased Consistent Entropy Estimator for the Special Bernoulli Measure

Let \(\Omega = A^N\) - be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\), \(N = {1,2,\dots} \), \[\rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} \] - a metric on \(\Omega\), and \(\mu\) - a probability measure on \(\Omega\). Let \(\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots, \boldsymbol{\xi_n}\) - be independent identically distributed points on \(\Omega\). We study the estimator \(\eta_n^{(k)}(\gamma)\) - of the reciprocal of the entropy \(1/h\), that are defined as. \[\eta_n^{(k)}(\gamma) = k \left(r_{n}^{(k)}(\gamma) - r_{n}^{(k+1)}(\gamma)\right),\] where \[r_n^{(k)}(\gamma) =\frac{1}{n+1}\sum_{j=0}^{n} \gamma\left(\min_{i:i \neq j} {^{(k)}} \rho(\boldsymbol{\xi_{i}}, \boldsymbol{\xi_{j}})\right),\] \(\min ^{(k)}\{X_1,\dots,X_N\}= X_k\), если \(X_1\leq X_2\leq \dots\leq X_N\). Number \(k\) and a function \(\gamma(t)\) - are auxiliary parameters. The main result of this paper isTheorem.Let \(m\) - be the Bernoulli measure with probabilities \(p_0,p_1>0\), \(p_0+p_1=1\), \(p_0=p_1^2\), then \(\forall eps>0\) some continuous function \(\gamma(t)\) such that \[\left|E\eta_n^{(k)}(\gamma) - \frac1h\right| <eps,\quad DD\eta_n^{(k)}(\gamma)\to 0,n\to \infty. \]

Characterizing asymptotic randomization in abelian cellular automata

Abelian cellular automata (CAs) are CAs which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images have weak*-convergence towards the uniform Bernoulli measure (the Haar measure in this setting). We are interested in structural phenomena, i.e., randomization for a wide class of initial measures (under some mixing hypotheses). First, we prove that an abelian CA randomizes in Cesàro mean if and only if it has no soliton, i.e., a non-zero finite configuration whose time evolution remains bounded in space. This characterization generalizes previously known sufficient conditions for abelian CAs with scalar or commuting coefficients. Second, we exhibit examples of strong randomizers, i.e., abelian CAs randomizing in simple convergence; this is the first proof of this behaviour to our knowledge. We show, however, that no CA with commuting coefficients can be strongly randomizing. Finally, we show that some abelian CAs achieve partial randomization without being randomizing: the distribution of short finite words tends to the uniform distribution up to some threshold, but this convergence fails for larger words. Again this phenomenon cannot happen for abelian CAs with commuting coefficients.

Approximation of Bernoulli measures for non-uniformly hyperbolic systems

An invariant measure is called a Bernoulli measure if the corresponding dynamics is isomorphic to a Bernoulli shift. We prove that for$C^{1+\unicode[STIX]{x1D6FC}}$diffeomorphisms any weak mixing hyperbolic measure could be approximated by Bernoulli measures. This also holds true for$C^{1}$diffeomorphisms preserving a weak mixing hyperbolic measure with respect to which the Oseledets decomposition is dominated.

Approximate transitivity of the ergodic action of the group of finite permutations of on

In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.