The Non-Archimedean Theory of Discrete Systems

In the paper, we study behavior of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behavior of the system w.r.t. variety of word transformations performed by the system: We call a system completely transitive if, given arbitrary pair $a,b$ of finite words that have equal lengths, the system $\mathfrak A$, while evolution during (discrete) time, at a certain moment transforms $a$ into $b$. To every system $\mathfrak A$, we put into a correspondence a family $\mathcal F_{\mathfrak A}$ of continuous maps of a suitable non-Archimedean metric space and show that the system is completely transitive if and only if the family $\mathcal F_{\mathfrak A}$ is ergodic w.r.t. the Haar measure; then we find easy-to-verify conditions the system must satisfy to be completely transitive. The theory can be applied to analyze behavior of straight-line computer programs (in particular, pseudo-random number generators that are used in cryptography and simulations) since basic CPU instructions (both numerical and logical) can be considered as continuous maps of a (non-Archimedean) metric space $\mathbb Z_2$ of 2-adic integers.Comment: The extended version of the talk given at MACIS-201

Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $$ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$

T-functions revisited: New criteria for bijectivity/transitivity

The paper presents new criteria for bijectivity/transitivity of T-functions and fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz $p$-adic functions and to study measure-preservation/ergodicity of these

DISKRETNOST IZAZIVA TALASE

In the paper, we show that matter waves can be derived from discreteness and causality. Namely we show that matter waves can naturally be ascribed to finite discrete causal systems, the Mealy automata having binary input/output which are bit sequences. If assign real numerical values (‘measured quantities’) to bit sequences, the waves arise as a correspondence between the numerical values of input sequences (‘impacts’) and output sequences (‘system-evoked responses’). We show that among all discrete causal systems with arbitrary (not necessarily binary) inputs/outputs, only the ones with binary input/output can be ascribed to matter waves ψ(x, t) = ei(kx−ωt).U ovom radu pokazujemo da talasi materije mogu biti izvedeni iz diskretnosti i kauzalnosti. Naime, pokazujemo da talasi materije mogu biti prirodno pripisani konačnim diskretnim kauzalnim sistemima, Mealy automatima kod kojih su ulaz/izlaz binarni nizovi bitova. Ako nizovi bitova imaju realne numeričke vrednosti (merljive veličine), tada talasi nastaju kao veza izmedju numeričkih vrednosti ulaznih nizova (izazova) i izlaznih nizova (odgovora sistema). Pokazujemo da od svih diskretnih kauzalnih sistema sa proizvoljnim (ne obavezno binarnim) ulazima/izlazima,samo onima sa binarnim ulazom/izlazom mogu se pripisati talasi materije ψ(x, t) = ei(kx−ωt)