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Let ff be a C1+εC^{1+\\varepsilon } diffeomorphism on a compact smooth surface with positive topological entropy hh. For every 0>δ>h0>\\delta >h, we construct an invariant Borel set EE and a countable Markov partition for the restriction of ff to EE in such a way that EE has full measure with respect to every ergodic invariant probability measure with entropy greater than δ\\delta. The following results follow: ff has at most countably many ergodic measures of maximal entropy (a conjecture of J. Buzzi), and in the case when ff is C∞C^\\infty, lim supn→∞e−nh#{x:fn(x)=x}>0\\limsup \\limits _{n\\to \\infty }e^{-n h}\\#\\{x:f^n(x)=x\\}>0 (a conjecture of A. Katok).

Physical exercise has many physical, psychological and social health benefits leading to improved life quality. This paper presents a robotic system developed as a personal coach for older adults aiming to motivate older adults to participate in physical activities. The robot instructs the participants, demonstrates the exercises and provides real-time corrective and positive feedback according to the participant’s performance as monitored by an RGB-D camera. Two robotic systems based on two different humanoid robots (Nao, toy-like and Poppy, mechanical-like) were developed and implemented using the Python programming language. Experimental studies with 32 older adults were conducted, to determine the preferable mode and timing of the feedback provided to the user to accommodate user preferences, motivate the users and improve their interaction with the system. Additionally, user preferences with regards to the two different humanoid robots used were explored. The results revealed that the system motivated the older adults to engage more in physical exercises. The type and timing of feedback influenced this engagement. Most of these older adults also perceived the system as very useful, easy to use, had a positive attitude towards the system and noted their intention to use it. Most users preferred the more mechanical looking robot (Poppy) over the toy-like robot (Nao).

Given a continuous linear cocycle$\\mathcal {A}$over a homeomorphism f of a compact metric space X , we investigate its set$\\mathcal {R}$of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set$\\mathcal {R}$is of first Baire category (i.e., meager) in X , unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbański (2001) who showed that this condition is sufficient.

We study the entropy and Lyapunov exponents of invariant measures μ for smooth surface diffeomorphisms f , as functions of ( f , μ ) . The main result is an inequality relating the discontinuities of these functions. One consequence is that for a C ∞ surface diffeomorphism, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.

We describe a method for proving subexponential lower bounds for correlations functions, and apply it to study decay of correlations for maps with countable Markov partitions. One result is that LS Young's upper estimates [Y2] are optimal in many situations. Our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on Banach spaces, which we apply to the iterates of the transfer operator.[PUBLICATION ABSTRACT]

We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential $\\chi ^2$form.

We classify the ergodic invariant Radon measures for horocycle flows on ^sup d^-covers of compact Riemannian surfaces of negative curvature, thus proving a conjecture of M. Babillot and F. Ledrappier. An important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary. [PUBLICATION ABSTRACT]

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams.

Given a continuous linear cocycle A over a homeomorphism f of a compact metric space X, we investigate its set R of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set R is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.