Isomorphisms and Topological Models

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate the joinings among entropy zero systems and prove the disjointness property among entropy zero systems using the dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.

Given a topological dynamical system that admits mistakes and an invariant measure, this paper proves the Brin-Katok formula in this case. Consequently, this paper shows the Brin-Katok formula for the mean Bowen ball which replacing the Bowen metrics with the mean Bowen metrics.

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems.We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter—Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.

Topological dynamics on nets

The aim of this course is to present methods coming from quasiconformal geometry in metric spaces which can be used to characterize conformal dynamical systems. We will focus on some specific classes of rational maps and of Kleinian groups (semi-hyperbolic rational maps and convex-cocompact Kleinian groups). These classes can be characterized among conformal dynamical systems by topological properties, which will enable us to define classes of dynamical systems on the sphere (coarse expanding conformal maps and uniform convergence groups). It turns out that these topological dynamical systems carry some non-trivial geometric information enabling us to associate a coarse conformal structure invariant by their dynamics. This conformal structure will be derived from hyperbolic geometry in the sense Gromov. We associate to this conformal structure a numerical invariant, the Ahlfors regular conformal dimension, which will contain the information that such topological dynamical systems are conjugate to genuine conformal dynamical systems.

This paper contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. Using the positivity result of symplectic displacement energy [4], we prove that any generator of a strong symplectic isotopy uniquely determine the latter. This yields a symplectic analogue of a result proved by Oh [12], and the converse of the main theorem found in [6]. Also, tools for defining and for studying the topological symplectic dynamical systems are provided: We construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all Hamiltonian homeomorphism is equivalent to Oh’s metric [12], define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. Several open conjectures are elaborated.

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the usual regional proximality relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have less intricate dynamics, and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions and a computation of the automorphism group of the minimal Robinson tiling.

We prove that any uniformly exponentially stable linear cocycle of matrices defined over a topological dynamical system can be reduced via suitable change of variables to a linear cocycle whose generator has a spectral norm less than 1 at each point. Moreover, we establish an analogous result for nonuniformly exponentially stable linear cocycles, i.e. cocycles with negative Lyapunov exponents. In addition, by using tools from ergodic theory, we obtain converse results that give conditions for (non)uniform exponential stability of linear cocyles under the assumption that they admit a suitable reduction to a linear cocycle whose generator has a spectral norm less than 1 at points that form a ‘large’ subset of our topological dynamical system in the appropriate sense. Finally, we discuss the versions of our results in the case of (non)uniformly hyperbolic linear cocyles.

On the Banach ⁎-algebra crossed product associated with a topological dynamical system

This thesis is dedicated to studying the theory of entropy and its relation to the Kolmogorov complexity. Originating in physics, the notion of entropy was introduced to mathematics by C. E. Shannon as a way of measuring the rate at which information is coming from a data source. There are, however, a few different ways of telling how much information there is: an alternative approach to quantifying the amount of information is the Kolmogorov complexity, which was proposed by A. N. Kolmogorov. The Shannon entropy is the key ingredient in the definition of the Kolmogorov-Sinai entropy of a measure-preserving systems. Roughly speaking, the Kolmogorov-Sinai entropy is the expected amount of information in `Shannon sense' that one obtains per unit of time by observing the evolution of a measure-preserving system. In topological dynamics, the topological entropy takes place of the Kolmogorov-Sinai entropy. For metrizable systems, the topological entropy measures the exponential growth rate of the number of distinguishable partial orbits of length n as n tends to infinity. Originally defined for Z-actions, the 'classical' theories of entropy were later extended to actions of amenable groups. We provide a necessary background on amenable groups, topological/measurepreserving dynamics and the entropy theory in Chapters 1, 2, 3 and 5. The main focus of this thesis is extending the following results. First of all, a common generalization of the topological and the Kolmogorov-Sinai entropy theories for Z-systems was suggested by G. Palm. We provide an abstract generalization of the work of Palm for actions of discrete amenable groups in the language of measurement functors in Chapter 6. Secondly, we investigate the connection of entropy and Kolmogorov complexity. Originally, the equality between the topological entropy and a certain quantity measuring maximal asymptotic Kolmogorov complexity of the trajectories was established by A. A. Brudno for subshifts over Z. Later, he proved the equality of the Kolmogorov-Sinai entropy and the asymptotic Kolmogorov complexity of almost every trajectory for ergodic subshifts over Z. We provide a generalization of these results as follows. Firstly, in Chapter 4 we give a background on computability and Kolmogorov complexity and, further, introduce computable Folner monotilings, which are central in our extensions of Brudno's results. We treat the 'first' and the 'second' theorems of Brudno in Chapter 7. The first theorem is generalized for subshifts over computable groups admitting computable Folner monotilings, while the second theorem is proved under the assertion of regularity of the monotiling, which we introduce in Chapter 7 as well.

Entropy sets are defined both topologically and for ameasure. The set of topological entropy sets isthe union of the sets of entropy sets for all invariant measures.For a topological system $(X,T)$ and an invariant measure $\mu$ on$(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of theset of all entropy sets (resp. $\mu$- entropy sets) in thehyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp.$h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp.$(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has aninvariant measure with full support and infinite topologicalentropy. Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp.$h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropysets) is a dense $G_\delta$ in$H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed. Concerning entropy capacities, it is shown that when $\mu$ is ergodicwith $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ suchthat the Bowen entropy $h(E)\ge h_\mu(T)$is residual in $H^\mu(X,T)$. When in addition$(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system ( f , X , Ω ) (f,X,\Omega ) . The main idea is to make use of, in addition to invariant compact subspaces of ( X , Ω ) (X,\Omega ) , compactifications of étale covers π : ( f ′ , X ′ , Ω ′ ) → ( f , X , Ω ) \pi :(f’,X’,\Omega ’)\rightarrow (f,X,\Omega ) ; that is, π ∘ f ′ = f ∘ π \pi \circ f’=f\circ \pi and the fibers of π \pi are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an étale analog of algebraic dynamical systems.

<title>Abstract</title> Malignant cells sustain growth and therapy resistance by overexpressing nutrient transporters (GLUT1/3, LAT1, ASCT2, MCT1/4) within aberrantly glycosylated membranes. These carriers assemble into a redundant influx–efflux lattice, such that inhibition of one pathway is rapidly bypassed, rendering conventional blockade ineffective. Here we adapt the principle of \emph{topological replica neutralization}, originally formulated in condensed-matter physics, to metabolic oncology. In topological systems, inversion of polarization generates a replica with opposite Euler class; superposition cancels global invariants and destabilizes protected modes. Inspired by this mechanism, we design a biochemical \emph{inverse replica} that neutralizes malignant redundancy at its structural root. The strategy employs a branched polyethylene glycol scaffold bearing sulfonates (Cap--PEG--SO$_3^-$). Sulfonate termini provide \emph{electrostatic anchoring} to Lys/Arg-rich membrane patches, while the PEG backbone enforces \emph{steric occlusion}, coherently reducing conductance across transporter families. Scaffold sulfonation was validated by FTIR, NMR, and mass spectrometry, and PEGylation ensures solubility and compatibility. Modeling shows that this dual mechanism enforces global contraction $V_{\max,i}\mapsto(1-\theta_i)V_{\max,i}$, driving $Q_{\mathrm{norm}}:1\!\to 0$—a biochemical realization of Euler-class cancellation. Within the spectral--topological model, redundancy appears as a meronic half-covering ($Q=\tfrac{1}{2}$) that resists local perturbations. Global capping trivializes $e(\mathcal{E}\oplus\mathcal{E}^\ast)=0$, annihilating the invariant stabilizing robustness. The framework predicts a critical occlusion $\theta_c$ at which $Q$ collapses discretely while flux decays smoothly, with elasticities $\mathcal{E}_{\theta_i}[J_{\mathrm{tot}}]\approx -w_i([S])$. Thus, malignant resilience is reframed as a \emph{geometric obstruction} rather than kinetic surplus. Linking characteristic-class mathematics with scaffold engineering, this work outlines a translational paradigm: \emph{inverse replicas for neutralizing pathological transport}.

The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps.