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Stefano Luzzatto

Publications and source records attributed to Stefano Luzzatto.

At least 19 recordsLinked to original sources
arXiv2025

Rigorous computation of expansion in one-dimensional dynamics

We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted proof). The method uses efficient graph algorithms and an iterative approach for optimal performance. A software implementation of the method is made publicly available. This is an example of a quantitative result in the theory of dynamical systems, as opposed to many qualitative results whose assumptions may be difficult to verify and the conclusions may have limited use in practical models that describe natural phenomena. We discuss and illustrate the effectiveness of our method and apply it to the quadratic map family.

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arXiv2025

Finite Time Hyperbolic Coordinates

We define finite-time hyperbolic coordinates, describe their geometry, and prove various results on both their convergence as the time scale increases, and on their variation in the state space. Hyperbolic coordinates reframe the classical paradigm of hyperbolicity: rather than define a hyperbolic dynamical system in terms of a splitting of the tangent space into stable and unstable subspaces, we define hyperbolicity in terms of the co-eccentricity of the map. The co-eccentricity describes the distortion of unit circles in the tangent space under the differential of the map. Finite-time hyperbolic coordinates have been used to demonstrate the existence of SRB measures for the Henon map; our eventual goal is to both elucidate these techniques and to extend them to a broad class of nonuniformly and singular hyperbolic systems.

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arXiv2023

Persistent Non-Statistical Dynamics in One-Dimensional Maps

We study a class $\widehat{\mathfrak{F}}$ of one-dimensional full branch maps introduced in [Doubly Intermittent Full Branch Maps with Critical Points and Singularities; D. Coates, S. Luzzatto, M. Mubarak, 2022], admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that $\widehat{\mathfrak{F}}$ can be partitioned into 3 pairwise disjoint subfamilies $$\widehat{\mathfrak{F}} = \mathfrak{F} \cup \mathfrak{F}_\pm \cup \mathfrak{F}_*$$ such that all $g \in \mathfrak{F}$ have a unique physical measure equivalent to Lebesgue, all $g \in \mathfrak{F}_{\pm}$ have a physical measure which is a Dirac-$\delta$ measure on one of the (repelling) fixed points, and all $g \in \mathfrak{F}_{*}$ are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies.

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arXiv2023

Statistical stability of interval maps with critical points and singularities

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy some expansivity and bounded recurrence conditions. This generalizes known results for maps with critical points and bounded derivatives and in particular proves statistical stability of Lorenz-like maps with critical points and singularities studied in [S. Luzzatto and W. Tucker. Non-uniformly expanding dynamics in maps with singularities and criticalities. Inst. Hautes Etudes Sci. Publ. Math., (89):179-226, 1999]. We introduce a natural metric on the space of maps with discontinuities which does not seem to have been used in the literature before.

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arXiv2022

Doubly Intermittent Full Branch Maps with Critical Points and Singularities

We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild assumptions we prove the existence of a unique invariant mixing absolutely continuous probability measures, study its rate of decay of correlation and prove a number of limit theorems.

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arXiv2021

Rigorous computation of escape times for parameter intervals in the quadratic map

We study the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We conduct comprehensive numerical analysis of collections of finite orbits of the critical point, computed for intervals of parameter values using rigorous numerical methods. We use the computer to explicitly construct a collection of several thousand parameter intervals, contained in $\Omega=[1.4, 2]$, that are proved to have a specific so-called escape time, which roughly means that some effectively computed iterate of the critical point taken over all the parameters in that interval has considerable width in the phase space. In particular, we compute a rigorous lower bound on this width, in addition to the upper bound. We investigate the effect of certain constraints imposed on the numerical computations upon the resulting collection of intervals. Additionally, we illustrate and discuss the distribution of the computed intervals in the parameter space. The purpose of our work is to establish grounds for further numerical computation of a lower bound on the measure of stochastic parameters in $\Omega$. The source code of the software and the data discussed in the paper are freely available at http://www.pawelpilarczyk.com/quadr/. This web page also allows carrying out some limited computations. The ideas and procedures introduced in the paper can be easily generalised to apply to other parametrised families of dynamical systems.

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arXiv2020

Rigorous numerics for critical orbits in the quadratic family

We develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We illustrate the effectiveness of our approach by constructing a dynamically defined partition $\mathcal P$ of the parameter interval $\Omega=[1.4, 2]$ into almost 4 million subintervals, for each of which we compute to high precision the orbits of the critical points up to some time $N$ and other dynamically relevant quantities, several of which can vary greatly, possibly spanning several orders of magnitude. We also subdivide $\mathcal P$ into a family $\mathcal P^{+}$ of intervals which we call stochastic intervals and a family $\mathcal P^{-}$ of intervals which we call regular intervals. We numerically prove that each interval $\omega \in \mathcal P^{+}$ has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in $\omega$ has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in $\mathcal P^{+}$ are stochastic and most parameters belonging to the intervals in $\mathcal P^{-}$ are regular, thus the names. We prove that the intervals in $\mathcal P^{+}$ occupy almost 90% of the total measure of $\Omega$. The software and the data is freely available at http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying out the calculations. The ideas and procedures can be easily generalized to apply to other parametrized families of dynamical systems.

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arXiv2019

SRB measures and Young towers for surface diffeomorphisms

We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+\alpha}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in nonuniformly hyperbolic sets which is of independent interest.

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arXiv2016

Integrability of Continuous Bundles

We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.

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arXiv2016

The geometric approach for constructing Sinai-Ruelle-Bowen measures

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.

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arXiv2015

Young Towers for Product Systems

We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.

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arXiv2015

Integrability of dominated decompositions on three-dimensional manifolds

We investigate the integrability of 2-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular we prove unique integrability of dynamically dominated and volume dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

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arXiv2015

Integrability of C^1 invariant splittings

We derive some new conditions for integrability of dynamically defined C^1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C^1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.

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arXiv2015

Uniform expansivity outside the critical neighborhood in the quadratic family

We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We compute a possibly small radius of the critical neighborhood, and a lower bound for the corresponding expansivity exponent outside this neighborhood, valid for all the parameters in each of the intervals. We illustrate and study the distribution of the radii and these exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/..

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arXiv2012

Geometry of expanding absolutely continuous invariant measures and the liftability problem

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.

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arXiv2011

Uniform hyperbolic approximations of measures with non zero Lyapunov exponents

We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Omega_n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets, such that for any sequence mu_n of f-invariant ergodic probability measures with supp (mu_n) in Omega_n we have mu_n -> mu in the weak-* topology.

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