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A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Holder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowens arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.
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Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $Lambda$ is center topologically mixing then $f|_{Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)leq h(f,mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
In the present paper, we study the distribution of the return points in the fibers for a RDS (random dynamical systems) nonuniformly expanding preserving an ergodic probability, we also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property between fibers. As consequences, we prove that any expanding measure is the limit of probability measure whose measures of disintegration on the fibers are supported by a finite number of return points and we prove that the average of the measures on the fibers corresponding to a disintegration, along an orbit in the base dynamics is the limit of Dirac measures supported in return orbits on the fibers.
A dynamical system is a pair $(X,G)$, where $X$ is a compact metrizable space and $G$ is a countable group acting by homeomorphisms of $X$. An endomorphism of $(X,G)$ is a continuous selfmap of $X$ which commutes with the action of $G$. One says that a dynamical system $(X,G)$ is surjunctive provided that every injective endomorphism of $(X,G)$ is surjective (and therefore is a homeomorphism). We show that when $G$ is sofic, every expansive dynamical system $(X,G)$ with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.
In systems biology modeling, important steps include model parameterization, uncertainty quantification, and evaluation of agreement with experimental observations. To help modelers perform these steps, we developed the software PyBioNetFit. PyBioNetFit is designed for parameterization, and also supports uncertainty quantification, checking models against known system properties, and solving design problems. PyBioNetFit introduces the Biological Property Specification Language (BPSL) for the formal declaration of system properties. BPSL allows qualitative data to be used alone or in combination with quantitative data for parameterization model checking, and design. PyBioNetFit performs parameterization with parallelized metaheuristic optimization algorithms (differential evolution, particle swarm optimization, scatter search) that work directly with existing model definition standards: BioNetGen Language (BNGL) and Systems Biology Markup Language (SBML). We demonstrate PyBioNetFits capabilities by solving 31 example problems, including the challenging problem of parameterizing a model of cell cycle control in yeast. We benchmark PyBioNetFits parallelization efficiency on computer clusters, using up to 288 cores. Finally, we demonstrate the model checking and design applications of PyBioNetFit and BPSL by analyzing a model of therapeutic interventions in autophagy signaling.
Let a countable amenable group $G$ act on a zd compact metric space $X$. For two clopen subsets $mathsf A$ and $mathsf B$ of $X$ we say that $mathsf A$ is emph{subequivalent} to $mathsf B$ (we write $mathsf Apreccurlyeq mathsf B$), if there exists a finite partition $mathsf A=bigcup_{i=1}^k mathsf A_i$ of $mathsf A$ into clopen sets and there are elements $g_1,g_2,dots,g_k$ in $G$ such that $g_1(mathsf A_1), g_2(mathsf A_2),dots, g_k(mathsf A_k)$ are disjoint subsets of $mathsf B$. We say that the action emph{admits comparison} if for any clopen sets $mathsf A, mathsf B$, the condition, that for every $G$-invariant probability measure $mu$ on $X$ we have the sharp inequality $mu(mathsf A)<mu(mathsf B)$, implies $mathsf Apreccurlyeq mathsf B$. Comparison has many desired consequences for the action, such as the existence of tilings with arbitrarily good F{o}lner properties, which are factors of the action. Also, the theory of symbolic extensions, known for $mathbb z$-actions, extends to actions which admit comparison. We also study a purely group-theoretic notion of comparison: if every action of $G$ on any zero-dimensional compact metric space admits comparison then we say that $G$ has the emph{comparison property}. Classical groups $mathbb z$ and $mathbb z^d$ enjoy the comparison property, but in the general case the problem remains open. In this paper we prove this property for groups whose every finitely generated subgroup has subexponential growth.
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