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For every generalized β-map τ introduced by Góra [P. Góra, Invariant densities for generalized β-maps, Ergod. Theory Dyn. Syst. 27 (2007), pp. 1583–1598], we find an explicit formula for a basis of the (generalized) eigenspace corresponding to an isolated eigenvalue of its Perron–Frobenius operator on the space of functions of bounded variation. From this formula, we see that any (generalized) eigenfunction is a singular function related to the orbit at 1 by the map τ. In addition, as a consecutive work of the paper [S. Suzuki, Artin-Mazur zeta functions of generalized β-transformations, Kyushu J. Math. 71 (2017), pp. 85–103], the analytic continuation of its lap-counting function is given by the generating function for the coefficient sequence of the τ-expansion of 1. [ABSTRACT FROM AUTHOR]

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator W τ on a reproducing kernel Hilbert space (RKHS). The operator W τ is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, W τ generates a unitary evolution group { e t W τ } t ∈ R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of W τ as τ → 0 + , as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented. [ABSTRACT FROM AUTHOR]

We establish equivalent conditions for the existence of an integrable or locally integrable fixed point for a Markov operator with the maximal support. Maximal support means that almost all initial points will concentrate on the support of the invariant density under the iteration of the process. One of the equivalent conditions for the existence of a locally integrable fixed point is weak almost periodicity of the jump operator with respect to some sweep-out set. This result includes the case of the existence of an absolutely continuous σ-finite invariant measure when we consider a nonsingular transformation on a probability space. Weak almost periodicity implies the Jacobs-Deleeuw-Glicksberg splitting and we show that constrictive Markov operators which guarantee the spectral decomposition are typical weakly almost periodic operators. [ABSTRACT FROM AUTHOR]

We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron–Frobenius groups of unitary operators in a smooth orthonormal basis of the L 2 space of the dynamical system, acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection–diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields that transform naturally under changes of observation modality, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also establish a correspondence between Koopman operators and Laplace–Beltrami operators constructed from data in Takens delay-coordinate space, and use this correspondence to provide an interpretation of diffusion-mapped delay coordinates for this class of systems. Moreover, we take advantage of a special property of the Koopman eigenfunction basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for probability densities and observables. In systems with more complex spectral behavior, including mixing systems, we develop a method inspired from time change in dynamical systems to transform the generator to a new operator with potentially improved spectral properties, and use that operator for vector field decomposition and nonparametric forecasting. [ABSTRACT FROM AUTHOR]

Let J be the set of all infinite complex continued fractions with partial numerators equal to one and partial denominators b 1 , b 2 , … , where each b k is a complex number of the form m + n i with m being a positive integer and n being any integer. In this paper we give an algorithm to determine lower bounds for the Hausdorff dimension of J . We show that the Hausdorff dimension of J is greater than 1.825, which is the best known lower bound, to the best of our knowledge. The ideas used to obtain this improved bound is different from those used by other authors and could be applied to estimate Hausdorff dimensions of other sets as well. [ABSTRACT FROM AUTHOR]

It was proved by Urbański and Zdunik (Fund Math 220:23-69, ) that for every holomorphic endomorphism $$f:{{\mathbb { P}}}^k\rightarrow {{\mathbb { P}}}^k$$ of a complex projective space $${{\mathbb { P}}}^k,k\ge 1$$ , there exists a positive number $$\kappa _f>0$$ such that if $$J$$ is the Julia set of $$f$$ (i.e. the support of the maximal entropy measure) and $$\phi :J\rightarrow {\mathbb R}$$ is a Hölder continuous function with $$\sup (\phi )-\inf (\phi )<\kappa _f$$ (pressure gap), then $$\phi $$ admits a unique equilibrium state $$\mu _\phi $$ on $$J$$ . In this paper we prove that the dynamical system ( $$f,\mu _\phi $$ ) enjoys exponential decay of correlations of Hölder continuous observables as well as the Central Limit Theorem and the Law of Iterated Logarithm for the class of these variables that, in addition, satisfy a natural co-boundary condition. We also show that the topological pressure function $$t\mapsto P(t\phi )$$ is real-analytic throughout the open set of all parameters $$t$$ for which the potentials $$t\phi $$ have pressure gaps. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron–Frobenius operators, invariant measures and representations of the Cuntz–Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz–Krieger algebras for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of . We use these representations to describe the Perron–Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples. [Copyright &y& Elsevier]

We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible, time-dependent fluid flows driven by ergodic dynamical systems. Our approach is based on representations of the generators of the Koopman and Perron–Frobenius groups of operators governing the evolution of observables and probability measures on Lagrangian tracers, respectively, in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm. These operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place, and as a result their eigenfunctions correspond to global space–time coherent patterns under a skew-product dynamical system. Moreover, using this data-driven representation of the generators in conjunction with Leja interpolation for matrix exponentiation, we construct model-free prediction schemes for the evolution of observables and probability densities defined on the tracers. We present applications to periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96 systems. • Lagrangian tracer dynamics studied via an extended-state space, skew-product approach. • Coherent spatiotemporal patterns via Koopman eigenfunctions for skew-product systems. • Predictive models constructed through exponentiation of the Koopman generator. • Data-driven formulation with convergence guarantees developed using kernel algorithms. • Framework applied to periodic vortex flows and aperiodic flows driven by L96 systems. [ABSTRACT FROM AUTHOR]