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This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps

We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

This paper deals with the problem of establishing a systematic theoretical formulation of variational principles for the continuum gravitational field dynamics of classical General Relativity (GR). In this reference, the existence of multiple Lagrangian functions underlying the Einstein field equations (EFE) but having different physical connotations is pointed out. Given validity of the Principle of Manifest Covariance (PMC), a set of corresponding variational principles can be constructed. These are classified in two categories, respectively, referred to as constrained and unconstrained Lagrangian principles. They differ for the normalization properties required to be satisfied by the variational fields with respect to the analogous conditions holding for the extremal fields. However, it is proved that only the unconstrained framework correctly reproduces EFE as extremal equations. Remarkably, the synchronous variational principle recently discovered belongs to this category. Instead, the constrained class can reproduce the Hilbert–Einstein formulation, although its validity demands unavoidably violation of PMC. In view of the mathematical structure of GR based on tensor representation and its conceptual meaning, it is therefore concluded that the unconstrained variational setting should be regarded as the natural and more fundamental framework for the establishment of the variational theory of EFE and the consequent formulation of consistent Hamiltonian and quantum gravity theories.

We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.

The modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations of complex crack topologies including branching. This drawback can be overcome by a diffusive crack modeling based on the introduction of a crack phase field as proposed in Miehe et al. (Comput Methods Appl Mech Eng 19:2765–2778, 2010a ; Int J Numer Meth Eng 83:1273–1311, 2010b ), Hofacker and Miehe (Int J Numer Meth Eng, 2012 ). In this work, we summarize basic ingredients of a thermodynamically consistent, variational-based model of diffusive crack propagation under quasi-static and dynamic conditions. It is shown that all coupled field equations, in particular the balance of momentum and the gradient-type evolution equation for the crack phase field, follow as the Euler equations of a mixed rate-type variational principle that includes the fracture driving force as the mixed field variable. This principle makes the proposed formulation extremely compact and provides a perfect basis for the finite element implementation. We then introduce a local history field that contains a maximum energetic crack source obtained in the deformation history. It drives the evolution of the crack phase field. This allows for the construction of an extremely robust operator split scheme that updates in a typical time step the history field, the crack phase field and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading.

Motivated by fractal geometry of self-affine carpets and sponges, Feng and Huang [J. Math. Pures Appl. 106(9) (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng and Huang. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford–McMullen carpet in purely topological terms.

The generalized Reissner-type operator of three-dimensional micropolar mechanics of solids is presented, on the basis of which the generalized Reissner-type operator of three-dimensional micropolar mechanics of thin solids with one small size is obtained under the new parameterization of the domains of these bodies. From the last Reissner-type operator, in turn, the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size is derived under the new parametrization of the domains of these bodies. It should be noted that the advantage of the new parameterization is that it is experimentally more accessible than other parameterizations (Nikabadze in Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies, MSU Publishing House, 2014; Contemp Math. Fundam Dir 55:3–194, 2015; J Math Sci 225:1, 2017). Further, applying the method of orthogonal polynomials (expansion of unknown quantities in series in terms of a system of orthogonal polynomials), from the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size under the new parameterization of the domains of these bodies, the Reissner variational principle of micropolar mechanics of thin solids with one small size in the moments with respect to the system of Legendre polynomials is derived. In addition, the method is described for obtaining the variational principles of Lagrange and Castigliano of micropolar mechanics of thin solid with one small size under the new parametrization of the domains of these bodies in moments with respect to systems of the first and second kind Chebyshev polynomials. The paper is a continuation of the work “Nikabadze, Ulukhanyan, On some variational principles in the three-dimensional micropolar theories of solid”; therefore, before reading this paper, the authors invite the interested reader to familiarize themselves with the work (Nikabadze and Ulukhanyan in On some variational principles in the three-dimensional micropolar theories of solids, submitted).

The effective parametrization of a multilayer thin domain, called a new parametrization, is considered and consists in using, in contrast to the classical approaches, several base surfaces. In addition, the new parameterization is characterized by the fact that it is experimentally more accessible than other parameterizations used in the scientific literature, since the front surfaces are used as basic ones. Also, when obtaining any relation (a system of equations, constitutive relations, boundary and initial conditions, variational principles, etc.) in the moments of the theory of multilayer thin bodies under the new parametrization of the domain of a thin body, it is sufficient in the corresponding relation of the theory of a single-layer thin body under the root letters of the quantities to supply the index α, which denotes the number of the layer α and gives these index values from 1 to K, where K is the number of layers. Therefore, for the correct statement of the initial-boundary value problems to the equations of motion and the boundary and initial conditions in the moments, it is also necessary to add interlayer contact conditions, which must also be taken into account when writing the variational operators and formulating the variational principles. What has been said above can be called the rule of obtaining the desired relation in the theory of multilayer thin bodies from the corresponding relation in the theory of single-layer thin bodies. Applying this rule, below we give the representation of the generalized Reissner-type operator and formulate the generalized Reissner-type variational principle both in the case of full contact of adjacent layers of a multilayer structure and in the presence of zones of weakened adhesion. The description of obtaining of dual operators and variational principles of Reissner-type, as well as of Lagrangian and Castiglianian and variational principles of Lagrange and Castigliano, is given. In the presence of domains of weakened adhesion at interphase boundaries in a multilayer thin body, one of the main problems is the problem of modeling the interface (interphase boundary). In this paper, the jump-type model (description of the interface by a surface of zero thickness) is presented in comparative detail.

In this article, we introduce a notion of relative mean metric dimension with potential for a factor map π : ( X , d , T ) → ( Y , S ) between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapira’s entropy, Katok’s entropy and Brin–Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi (On variational principles for metric mean dimension, 2021. arXiv:2101.02610 ) partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of ( Y ,  S ) are also investigated.

In this paper, we studied the metric mean dimension in Feldman–Katok (FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subsets. And we established two variational principles.